For the eclipse to happen, the two stars and two planets involved would have to orbit in the same plane. It is possible that they don't orbit in the same plane. But the two different orbital planes will have to intersect, and the eclipse could only happen when the when the two planets and the K star are lined up in the intersection of the two planes.
Thus most times when the two planets align with the K type star, their planes will be tilted and the eclipse shadow will pass "over" or "below" the planet. It will be very rare for the two planets to align with the K type star when the two planets and the star are in the line where the two orbital planes intersect. Thus by changing the parameters such as the orbital periods and distances of various objects and the relative tilt of the two planes, it should be possible to make an eclipse of the K type star by planet 2 as seen on the main planet extremely rare.
With the two planets each orbiting one of the two stars in a non-circumbinary or S-type orbit, there are some limits on how close the two stars can get to each other.
In non-circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.1
In non-circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.1
https://en.wikipedia.org/wiki/Habitability_of_binary_star_systems#Non-circumbinary_planet_(S-Type)
In July 2010, some astronomers estimated that 44 percent of F6 to K3 of the main sequence stars in the solar neighborhood that are possibly suitable (i.e., with a stellar mass between 1.5 and 0.5 times that of Sol) for hosting Earth-type planets may be members of binary or multiple star systems, possibly declining to one third to one fourth of very dim type M stars fsthat are difficult to observe (Raghavan et al, 2010; Charles J. Lada, 2006; and Duquennoy and Mayor, 1991). In binary star systems, however, a planet must not be located too far away from either one star or too close to two "home" stars or its orbit will be unstable. If that distance exceeds about one fifth of the closest approach of the other star, then the gravitational pull of that second star can disrupt the orbit of the planet (Graziani and Black, 1981; Pendleton and Black, 1983; and Dvorak et al, 1989). Indeed, stable orbits may extend as far as one third of the closest separation between any two stars in a binary system, but according to NASA's Kepler Mission team, numerical integration models have shown that there is a range of orbital radii between about 1/3 and 3.5 times the stellar separation for which stable orbits around two stars are not possible (Holman and Wiegert, 1999; Wiegert and Holman, 1997; and Donnison and Mikulskis, 1992). In star systems with more than two stars, the limits on stable orbital distance are so stringent that the presence of Earth-type planets in habitable orbits where surface water would be liquid are much less likely.
http://www.solstation.com/habitable.htm#sthash.oBAG0maF.dpbs
So the distance between the two stars must be at least five times the distance at which planet 2 orbits the K type star, and at least five times the distance that the main star orbits the G type star.
You say that the main planet orbits the G type star at a distance of one Au with a period of about 275 days. Since the Earth orbits a G type star at a distance of one AU with a period of 365.25 days, and since the Sun is believed to be G2V, it is unlikely that even the most massive main sequence G-type star could have an orbital period as short as 275 days at a distance of 1 AU. Thus you may need to change the orbital distance and/or period.
Let's assume that the main planet orbits around a star with spectral type ranging from G0V to G9V.
A G0V star would have a mass of 1.06 the Sun's mass and a luminosity of 1.35 the Sun's luminosity.
A G9V star would have a mass of 0.90 the Sun's mass and a luminosity of 0.55 the Sun's luminosity.
https://en.wikipedia.org/wiki/G-type_main-sequence_star#:~:text=A%20G%2Dtype%20main%2Dsequence,about%205%2C300%20and%206%2C000%20K.
The distance of what I call the Earth Equivalent Distance, or EED, of a star, the distance where a planet would receive exactly as much radiation from the star as Earth Gets from the Sun, is 1 AU, the distance of Earth from the Sun, multiplied by the square root of the star's luminosity divided by that of the Sun.
Thus the G0V star would have a EED at 1.1619 AU, and a G9V star would have an EED at 0.7416 AU.
A planet at the EED of a G0V star would have an orbital period of 1.222 Earth years, and a planet at the EED of a G9V star would have an orbital period of 0.763 Earth years.
Of course a habitable planet doesn't have to orbit at exactly the EED distance of its star, but could orbit closer or farther if it was within the star's circumstellar habitable zone.
There is a real simple method to find the inner and outer degrees of a star's circumstellar habitable zone. Just find the inner and outer edges of the Sun's circumstellar habitable zone and multiply them by the square root of the star's luminosity divided by the Sun's luminosity.
Here is a link to a list of 15 estimates of one or both edges of the Sun's circumstellar habitable zone made in the last 60 years.
https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates
They do not all agree very closely. So my advice for a science fiction writer who only wants to have one habitable planet in a fictional star system is that they should put that planet within one percent of the EED of the star.
Anyway, unless unless you want to turn the G type star into a F type or K type star, the main planet will orbit the G type star at a distance somewhere between about 0.7416 and 1.1619 AU. Which means that the minimum possible separation between the two stars should be at least 3.708 to 5.8095 AU.
The minimum possible separation between the two stars will have to be even greater if planet 2 orbits the K-type star at a greater distance than the main planet orbits the G-type star. If planet 2 orbits at 5 AU, the separation between the stars will have to be at least 25 AU, if it orbits at 10 AU the separation between the stars will have to be at least 50 AU, and so on.
Then there is problem that a typical star has many time the diameter of even the largest possible planet. Because most planets are much smaller than most stars the planet will usually appear as black dot against the right background of the star instead of making a total eclipse of the star.
When the planet is smaller than the star it passes in front of, which is almost always the case, the shadow of the planet that a total eclipse of the star can be seen will get narrower and narrower with increasing distance from the planet until it will come to a point at some distance, beyond which a total eclipse will not be visible.
If, however, the planet has a greater diameter than the star it passes in front of, the shadow of the planet within which a total eclipse of the star can been will get wider and wider with increasing distance from the planet and will extend infinitely far into space.
But planet's don't get much larger than Jupiter, even though they get several times as massive as Jupiter. Planets much more massive than Jupiter get really compressed by their gravity and become denser and denser. Thus planets only get a little bigger than Jupiter, and more massive planets become smaller and smaller. There is only one exception to this rule, which I will mention later. This also applies to the objects call brown dwarfs which have masses between about 13 Jupiter masses and about 80 times the mass of Jupiter.
In fact, I have read that planets more massive than Jupiter, brown dwarfs, and the least massive stars all tend to have diameters within 15 percent of he diameter of Jupiter.
And that offers some hope. If you make the K-type star a very low mass spectral type M star instead, it is possible that it would have a diameter similar to those of the widest planets.
According to this article from 2009:
The smallest known star right now is OGLE-TR-122b, a red dwarf star that’s part of a binary stellar system. This red dwarf the smallest star to ever have its radius accurately measured; 0.12 solar radii. This works out to be 167,000 km. That’s only 20% larger than Jupiter. You might be surprised to know that OGLE-TR-122b has 100 times the mass of Jupiter, but it’s only a little larger.
https://www.universetoday.com/25348/what-is-the-smallest-star/
According to this article there is a star EBLM Jo555-57 Ab which is smaller in diameter than Saturn.
https://tech.hindustantimes.com/tech/news/fire-in-the-sky-these-are-the-largest-and-smallest-stars-in-the-universe-136a1-tops-list-71664273467664.html#:~:text=Finally%2C%20the%20award%20for%20the,barely%20qualifies%20as%20a%20star.
EBLM J0555-57Ab has a mass of about 85.2±4 Jupiter masses, or 0.081 solar masses. Its radius is 0.08 solar radii (about 59,000 km), comparable to Saturn, which has an equatorial radius of 60,268 km. The star is about 250 times more massive than Saturn.1 Current stellar models put its mass at the lower limit for hydrogen-burning stars.
https://en.wikipedia.org/wiki/EBLM_J0555-57
The planet Jupiter has a mean radius of 69,911 kilometers.
This list include two other class M stars with radii less than that of Jupiter, SSSPM Jo829-1309 and SCR 184-63r7 A.
https://en.wikipedia.org/wiki/List_of_smallest_stars
A planet 10 percent wider than Jupiter would have a mean radius of 76,902.1 kilometers. And there are five more stars on the list with radii less than that.
So it is possible for extremely wide planets to be wider than the smallest red dwarf stars, even though those stars are many times as massive as any planet.
Thus you should consider replacing the K type star with a very small class M star, and making planet 2 a very large planet, so the planet will be wider than the star and cast an ever widening shadow into space. That will enable planet 2 to cast a total eclipse on the main planet at any distance when they are lined up correctly.
There is one way to get an even smaller star which will be smaller than a lot more normal sized planets.
Make the small star a white dwarf star. The largest white dwarf has a radius off 1874,530 kilometers, while the smallest white dwarf stars are smaller than the Earth. The smallest known white dwarf star, HD 49798, has a radius of 1,600 kilometers. A few white dwarfs are known to have planets orbiting them.
https://en.wikipedia.org/wiki/White_dwarf#Debris_disks_and_planets
However, all white dwarfs are former main sequence stars which became red giants at least once after their main sequence phases. So the closer the white dwarf is to the G type star and the main planet, the more likely it will be that it made the main planet uninhabitable when it was a red giant.
And there is one type of planet which can become much larger than Jupiter.
Hot Jupiters are gas giant planets orbiting very close to their stars and getting very hot.
Gas giants with a large radius and very low density are sometimes called "puffy planets"[48] or "hot Saturns", due to their density being similar to Saturn's. Puffy planets orbit close to their stars so that the intense heat from the star combined with internal heating within the planet will help inflate the atmosphere. Six large-radius low-density planets have been detected by the transit method. In order of discovery they are: HAT-P-1b,[49][50] COROT-1b, TrES-4, WASP-12b, WASP-17b, and Kepler-7b. Some hot Jupiters detected by the radial-velocity method may be puffy planets. Most of these planets are around or below Jupiter mass as more massive planets have stronger gravity keeping them at roughly Jupiter's size. Indeed, hot Jupiters with masses below Jupiter, and temperatures above 1800 Kelvin, are so inflated and puffed out that they are all on unstable evolutionary paths which eventually lead to Roche-Lobe overflow and the evaporation and loss of the planet's atmosphere.[51]
Even when taking surface heating from the star into account, many transiting hot Jupiters have a larger radius than expected. This could be caused by the interaction between atmospheric winds and the planet's magnetosphere creating an electric current through the planet that heats it up, causing it to expand. The hotter the planet, the greater the atmospheric ionization, and thus the greater the magnitude of the interaction and the larger the electric current, leading to more heating and expansion of the planet. This theory matches the observation that planetary temperature is correlated with inflated planetary radii.[51]
https://en.wikipedia.org/wiki/Hot_Jupiter#Puffy_planets
The largest known exoplanet is probably TYC 8998-760-1 b, with about 3 times the radius of Jupiter, a radius of abut 209,000 kilometers.
So I guess that if planet 2 orbits a very small class M star and is a very large planet, possibly a puffy planet close to the star, it should be able to cast a shadow all the way to the main planet and cause total eclipses of the small M star.
