I think that it is theoretically possible for a real world to have very similar climate, warm at the poles and cold at the equator. Unfortunately I can't do climate simulations for you but maybe you can find someone who can.
There is speculation that Earth might have been almost completely covered with ice and snow for millions of years hundreds of millions of years ago, so it is theoretically possible for Earth's tropics to have a very cold climate.
The problem is getting the planet's tropics cold at the same time the poles are warm. And my solution will make it impossible for your readers to ignore that the setting is on an alien planet in a different solar system. But fantasy stories can be set on alien planets in outer space just as well as on Earth or on flat worlds or in strange settings that are never explained, that have so much beyond the edges of the maps that the readers don't know if they are on spherical planets or flat discs.
You may have heard about the habitable zone around a star, a zone in which an Earth-like planet, if orbiting there, would receive the right amount of light and radiation from the star to have temperatures necessary for life.
Stars much more massive and brighter than Earth's sun would burn off all their hydrogen fuel much to fast for their planets to become habitable for humans or for intelligent life to evolve. Fortunately, most stars are red dwarf stars that can last for trillions of years. Unfortunately, the habitable zones around dim red dwarf stars are so close to the stars that planets orbiting in them would become tidally locked to their stars.
Their days would equal their years in length. One side of such a planet would always face toward the star, and be in eternal day and heat, and the other side would always face away from the star and be in eternal night and cold. And astrobiologists wondered and calculated whether life would be possible in such a world. So nobody knows if life bearing planets can orbit in the habitable zones of red dwarf stars.
When planets were first detected orbiting other stars, the first ones were very large, often several times the mass of Jupiter, and orbited close to their stars, often close enough to be roasted by the intense heat and light of their stars. Such worlds became known as "hot Jupiters".
And astrobiologists realized that if a "hot Jupiter" orbited in the habitable zone of a dim red dwarf, all its moons, if any, would be tidally locked to the "hot Jupiter" that they orbited and not to the star that the "hot Jupiter" orbited. Thus they would not keep one face always pointed at the star. Instead they would keep one face always pointed at the "hot Jupiter" and would have more more less normal days and nights.
And it is possible that some moons orbiting "hot Jupiters" could be several times as large as the largest moons in our solar systems and thus large enough to be habitable planets and have life. Thus it is possible that planet sized habitable exomoons orbiting "hot Jupiters" that orbit red dwarf stars might be as common as habitable planets orbiting other stars.
And this is important because I don't know what the size of the orbit of my hypothetical planet with warmer poles and colder equator will have to be. It is very possible that it will need to have a much shorter year than Earth and have to orbit very close around a red dwarf star, and then it will have to be a habitable exomoon of a giant planet instead of a habitable planet itself, in order to avoid being tidally locked to its star.
Most planets in our solar system spin with an axis of rotation that is almost perpendicular (at a right angle or 90 degrees) to the plane in which they orbit the sun. The rotational poles of six planets are inclined between 0.0 degrees (Mercury) and 28.8 degrees (Neptune) from such an expected right angle position. And most moons in the solar system have their axis of rotation ninety degrees from their orbital panes.
Thus it is expected that most extra solar planets will rotate around poles that are close to perpendicular to their orbital planes. But there will be some exceptions, like the two planets in our solar system that don't have such rotational poles. Venus rotates with an inclination of 177 degrees, almost exactly backwards from the normal position, and Uranus rotates with an inclination of 97 degrees, thus rotating almost exactly in its orbital plane.
Thus Uranus has very odd seasons in its year of 84.01 Earth years. Not that such odd seasons would matter very much on such a cold gas giant planet and its moons. But they would matter a lot on a planet in the habitable zone that had a axis of rotation inclined a similar amount.
Imagine a habitable planet orbiting in the habitable zone of its star, with an axis of rotation inclined about 90 degrees and thus almost in the plane that the planet orbits its star in.
In season A, the Northern hemisphere might be aimed almost exactly at its star. Thus the northern hemisphere would be in constant daylight and would be heating up all the time, especially the polar regions that would get the light coming almost straight down. The equatorial regions would not get very intense sunlight since it would be coming in very lo almost parallel to the ground, and any elevations would cast very long cold shadows.
The southern hemisphere would be cooling off in constant nighttime. Any people in the constant darkness would be able to watch the stars rotate 360 degrees every full rotation of the planet, unlike the natives of the northern hemisphere.
Season B. The planet is 90 degrees in its orbit from season A. Autumn in the northern hemisphere and spring in the southern hemisphere. Now the equatorial regions would face directly toward the star during the day and directly away during the night. Both the northern and southern hemispheres would also have alternating days and nights. The northern hemisphere would cool off and the southern hemisphere would warm up. Everybody anywhere on the planet could tell time by the position of the sun during the day and the position of stars and constellations during the night.
if the equatorial regions had a high altitudes and thin air and perhaps snow and ice on the ground to reflect light back into space, they might not warm up very much during that period.
Season C. 180 degrees of orbit from season A. The exact opposite of Season A. Northern hemisphere winter and constant night, and southern hemisphere summer and constant day. The equatorial regions get light at very low angles that doesn't heat them up very much, this time coming from the southern side and not the northern side.
Season D. 270 degrees of orbit from season A. The exact opposite of season B. spring in the warming northern hemisphere and autumn in the cooling southern hemisphere.
Now the equatorial regions would face directly toward the star during the day and directly away during the night. Both the northern and southern hemispheres would also have alternating days and nights. The northern hemisphere would warm up and the southern hemisphere would cool down. Everybody anywhere on the planet could tell time by the position of the sun during the day and the position of stars and constellations during the night.
And there might be intermediate seasons of change. Season AB between A and B, Season BC between B and C, Season CD between C and D, and season DA between D and A. During those seasons every part of the planet would get at least a little daytime and at least a little nighttime.
If the planet orbited around a fairly large and bright star like the Sun, each of the eight seasons might last for at least one Earth month.
But if the planet's year lasts as long as a Martian year of 1.88 Earth years, or an Earth year, or even a Venusian year of 0.62 Earth years, the summers at the poles might get too hot, and the winters at the poles might get too cold. You seem to want both poles to stay warmer than the equator all year round.
The polar temperature extremes can be moderated by ocean currents and winds carrying heat from warmer regions to cooler regions. But on Earth that is not enough to prevent many regions from having large temperature swings during the different seasons.
Thus the eight suggested seasons on that planet should be very short to prevent extreme temperature rises and falls at the poles. I guess that each season might last about two to four days of the planet, making each year last for about sixteen to thirty two days of the planet.
And with such a short orbit the planet will need to be an exomoon orbiting a hot, or at least warm, Jupiter-like planet. If it is a lone planet the tidal forces from the nearby star will gradually change its axis of rotation until it is almost at a perpendicular 90 degree angle to the planet's orbital plane. And it will slow down the planets rotation until the planet's day is the same length as its year, and then lock the rotation period. This will take mere millions of years early in the planet's history long before the first single celled life forms develop.
Unless the planet is an exomoon and will be tidally locked and protected from the star's tides by the planet it orbits - and thus keeps it odd axis of rotation until intelligent life develops, just as the moons of Uranus are locked into Uranus's axial tilt.
The article "Exomoon Habitability Constrained by Illumination and Tidal Heating" discusses the factors that affect the habitability of hypothetical exomoons.
The longest possible length of a satellite's day compatible with Hill stability has been shown to be about Pp/9, Pp being the planet's orbital period about the star (Kipping, 2009a).
Thus it is believed that the year of the planet and its habitable exomoon as they orbit their star will have to be at least nine times as long as the exomoon's day and month (or day/month. Since I suggested that the exomoon may have eight seasons, if they are equally long they will each last at least 1.125 of the exomoon's day/months.
Another possible advantage of an exomoon with a Uranus like inclination of its rotational axis is that tidal heating of an exomoon can keep a pole region warm during its long winter.
On the other hand, we can imagine scenarios where a moon becomes habitable only because of tidal heating. If the host planet has an obliquity similar to that of Uranus, then one polar region will not be illuminated for half the orbit around the star. Moderate tidal heating of some tens of watts per square meter might be just adequate to prevent the atmosphere from freezing out. Or if the planet and its moon orbit their host star somewhat beyond the outer edge of the IHZ, then tidal heating might be necessary to make the moon habitable in the first place. Tidal heating could also drive long-lived plate tectonics, thereby enhancing the moon's habitability (Jackson et al., 2008). An example is given by Jupiter's moon Europa, where insolation is weak but tides provide enough heat to sustain a subsurface ocean of liquid water (Greenberg et al., 1998; Schmidt et al., 2011). On the downside, too much tidal heating can render the body uninhabitable due to enhanced volcanic activity, as it is observed on Io.
The synchronized rotation periods of putative Earth-mass exomoons around giant planets could be in the same range as the orbital periods of the Galilean moons around Jupiter (1.7–16.7 d) and as Titan's orbital period around Saturn (≈16 d) (NASA/JPL planetary satellite ephemerides)4.
The authors don't list the length of an exomoon's day/month as a factor influencing its habitability, so for the moment we might assume that a habitable exomoon might have a rotational period or day/month of about 1.7 to 16.7 Earth days.
Thus the year of a habitable exomoon, that should be at least 9 times as long as its day/month, might be at least 15.3 to 150.3 Earth days long. But probably shorter than the 224.7 Earth days of Venus or even the 88.0 days of Mercury.
I suggested that each of the eight seasons of the exomoon might last for two to four days of the exomoon, and thus the total year could be about 27.2 to 534.4 Earth days.
Thus the year of the exomoon should be about 27.2 to 150.3 Earth days, and probably in the shorter part of that range.
The natives of the side of the exomoon that faces the planet should have a great view of the planet and any inner moons or rings it may have. Starlight reflected from the planet should illuminate their side very well and might even make it significantly warmer than the other side.
The natives of the far side of the exomoon might not even be aware that the planet exists.
Iapetus, a much smaller moon of Saturn than the hypothetical exomoon has a large equatorial bulge, with dimensions of 746 by 746 by 712 kilometers, and an equatorial ridge running three quarters around the moon making it look sort of like a walnut. The equatorial ridge is about 20 kilometers (12.42 miles) wide and 13 kilometers (8.07 miles) high.
If the exomoon had such a tall eqatorial ridge winds carrying warm moist air toward it during season A and season C would rain and/or snow on it's slopes. If the ridge was surrounded by tall plateaus like Tibet the precipitation would be snow that might pile up and turn into ice. thus the equatorial regions could be full of glaciers and ice sheets that reflected the light from the star back into space and never warmed up.