Have you see pictures or videos where Earth's Moon looks huge?
I believe that the angular diameter of the Moon, about half a degree of arc, is the same as a dime held at arm's length, which is pretty small. But the Moon looks vast in many pictures and videos because telephoto lenses are used to film it.
If you desire that both moons look like discs (when full) or partial discs (in other phases), both moons will have to be large enough to be (roughly) spherical.
Solar System objects more massive than 10 to the 21st power kilograms (one yottagram [Yg]) are known or expected to be approximately spherical. Astronomical bodies relax into rounded shapes (ellipsoids), achieving hydrostatic equilibrium, when their own gravity is sufficient to overcome the structural strength of their material. It was believed that the cutoff for round objects is somewhere between 100 km and 200 km in radius if they have a large amount of ice in their makeup;1 however, later studies revealed that icy satellites as large as Iapetus (1,470 kilometers in diameter) are not in hydrostatic equilibrium at this time,2 and a 2019 assessment suggests that many TNOs in the size range of 400-1000 kilometers may not even be fully solid bodies, much less gravitationally rounded.3 Objects that are ellipsoids due to their own gravity are here generally referred to as being "round", whether or not they are actually in equilibrium today, while objects that are clearly not ellipsoidal are referred to as being "irregular".
It was once expected that any icy body larger than approximately 200 km in radius was likely to be in hydrostatic equilibrium (HE).4 However, Ceres (r = 470 km) is the smallest body for which detailed measurements are consistent with hydrostatic equilibrium,8 whereas Iapetus (r = 735 km) is the largest icy body that has been found to not be in hydrostatic equilibrium. Earth's moon (r = 1,737 km) is also not in hydrostatic equilibrium, but—unlike icy Ceres and Iapetus—it is composed primarily of silicate rock, which has a much higher tensile strength than ice.
Vesta, which has a radius of 262.7 kilometers (166.3 miles), is the largest solar system object that looks irregular in a good photograph, although some of the ones larger than Vesta appear just as dots of light in the photograph.
More or less arbitrarily assuming that a radius of 400 kilometers is necessary for a rocky body to appear round, one can calculate how close such a small moon would have to be to appear as a disc and not a dot of light, in the sky of your world.
The maximum angular resolution of the human eye is 28 arc seconds or 0.47 arc minutes, this gives an angular resolution of 0.008 degrees, and at a distance of 1 km corresponds to 136 mm. This is equal to 0.94 arc minutes per line pair (one white and one black line), or 0.016 degrees. For a pixel pair (one white and one black pixel) this gives a pixel density of 128 pixels per degree (PPD).
So an object with a radius of 400 kilometers and a diameter of 800 kilometers would appear to be a tiny disc instead of a point of light if it had an angular diameter of at least 0.008 degrees of arc.
According to my rough calculations, that means that a minimum size round moon, with a radius or 400 kilometers (248.5 miles) and a diameter of 800 kilometers (497 miles), would have to be less than roughly 5,729,582.7 kilometers, or 3,560,197.6 miles, distant to be seen as a tiny disc and not as a mere point of light.
The Roche limit of an an astronomical body is the distance at which it will cause a smaller astronomical body to break up. For Earth the Roche limit is 9,492 kilometers (5,898 miles).
The Hill sphere of a planet, calculated from the masses of the planet and its star, and the distance between them, is the minimum distance that moon of the planet would have to be closer than in order stay in orbit around that planet.
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated.3
In the Earth-Sun example, the Earth (5.97×1024 kg) orbits the Sun (1.99×1030 kg) at a distance of 149.6 million km, or one astronomical unit (AU). The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU). The Moon's orbit, at a distance of 0.384 million km from Earth, is comfortably within the gravitational sphere of influence of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. All stable satellites of the Earth (those within the Earth's Hill sphere) must have an orbital period shorter than seven months.
So Earth's Hill sphere extends to about 1,500,000 kilometers (932,056.7 miles), and its true region of stability extends to about 500,000 to 750,000 kilometers, (310,685.5 to 466,028.3 miles).
The semi-major axis of the orbit of the Moon is 384,399 kilometers, or 238,854.4 miles.
So this means that if a habitable planet has similar mass and distance from its star (a star that should have a mass similar to that of the Sun) any moons which it has which are large enough to be round will be close enough to the planet to always appear round (except or their phases), and never appear as mere dots in the sky.
The minimum size of a rocky moon large enough to be round, with a radius or 400 kilometers (248.5 miles) and a diameter of 800 kilometers (497 miles), would be about 0.230229 of the diameter of the Moon, and thus about 0.0124228 of the volume of the Moon. If that moon had the same average density as the Moon, it would have about 0.0124228 of the mass of the Moon.
The gravitational pull of astronomical bodies one each other are proportional to their masses and distances. So if a smallest possible round moon was at the distance of Earth's moon, it would have only 0.0124228 as much gravitational attraction on Earth as the Moon does. According to my rough calculations, if a moon with a mass of 0.0124228 that of the Moon was at a distance of 0.1114576 of the Moon's distance, it would have a gravitational attraction on Earth equal to that of the Moon. That distance would be about 42,844.189 kilometers, or about 26,623.144 miles.
At that distance the minimum size round moon should appear to be roughly one arc degree wide, about twice the angular diameter of the Moon.
Thus my rough calculations indicate that it should be possible for a moon smaller than Earth's moon to be close enough to an Earth like planet to appear as large or larger than the Moon does from Earth without raising any higher tides.
Of course an astronomical situation which could possibly exist, is not the same thing as an astronomical situation which could form naturally, and an astronomical situation which could form naturally is not necessarily the same thing as an astronomical situation which could exist for the billions of years necessary until a planet developed an atmosphere with a lot of free oxygen and became habitable for beings with requirements similar to those of humans.
The Moon is believed to have formed out of debris much closer to Earth than it is now, and to have raised large tides on ancient Earth, and gradually slowed down the rotation of Earth and moved farther and farther from Earth.
The interactions between the two moons you desire, the planet, and its star could possibly eject one of the moons out of orbit around the planet long before the planet develops advanced multi celled life forms.
One possibility you might want to consider is making your "planet" a giant habitable moon of a giant planet the size of Saturn or Jupiter. The larger "moon" in the sky could be the giant planet, and the smaller "moon" could be another moon of the giant planet.
Rene Heller and jorge I. Zuluaga in "Magnetic Shielding of exomoons Beyond the Circumplanetary Edge" The Astrophysical Journal Letters, Volume 776, Issue 2, article id. L33, 6 pp. (2013) calculated the distances from a giant planet that a giant moon could potentially be habitable in. According to their calculations, the exomoon would have to be between 5 and 20 planetary radii to be shielded from radiation.
At a distance of 5 to 20 planetary radii, the planet would appear to be about 5.7295 to 22.9183 degrees wide, about 11 to 45 times the angular diameter of the Moon.
Any other moons of the giant planet that were large enough to be round would appear as discs and not dots whenever they were closer than roughly 5,729,582.7 kilometers, or 3,560,197.6 miles. Larger rounded moons would appear as discs at even larger distances.