Is a large moon necessary to stabilize the axial tilt of an Earth-like planet and keep it habitable?
This article suggests that the lack of a large moon would not make a planet uninhabitable:
This article suggests that the super high tides when the Moon was much closer to Earth may have caused life to evolve in vast tidal flats. Thus a large close moon might have been necessary for life to appear.
This article also suggests that the early Moon started plate tectonics and plate tectonics may be necessary for life on Earth.
So a large close moon early in a planet's life may or may not have been necessary for the planet to be habitable and for life to form. And a large moon much farther out billions of years later may or may not be necessary for the planet to have stable enough climate for life.
So turn your attention to the planet's atmosphere for a moment.
Earth's atmosphere scatters sunlight making it appear to come from every direction and making the sky appear blue. An object has to be brighter than the sky's background brightness to be visible in broad daylight. And during twilight it has to be brighter than the twilight sky's background to be visible.
In the apparent magnitude scale lower numbers signify greater apparent magnitude. It is said that the objects visible when the sun is less than 10 degrees above the horizon have to have an apparent magnitude of -2.5, while the faintest objects visible while the Sun is high in the sky have to have an apparent magnitude of -4.0. The apparent magnitude of the Moon varies from -12.90 when full down to -2.50 when new.
Since the new moon is seen when very close to the sun in the sky, it should never been visible unless the Sun is about to set or has already set, and so should hardly ever be visible when the Sun is. The much brighter full moon should always be opposite in direction to the Sun and thus only been seen at night.
So if the sky scatters enough light to become a few magnitudes brighter, its scattered light should be enough to drown out the light of the moon and make it invisible. Right?
I remember late one afternoon I was near Convention Hall, on the boardwalk of Cape May, New Jersey, and saw what looked like a full moon, as far as I could tell with the naked eye, rising low over the Atlantic Ocean.
And then I thought that a full looking Moon should be nearly 180 degrees from the Sun. But I was seeing a full looking Moon, close enough to full that I couldn't see the difference, not in the night sky, not even in twilight, but in the afternoon in broad daylight with blue sky.
So I turned around and looked back, and there was a red setting Sun low in the sky, opposite to the Moon. Because the Earth is a sphere, and not flat, the ground falls away in the distance, and one can sometimes see objects more than 90 degrees lower than the zenith of the sky. And atmosphere refraction makes objects near the horizon appear to be several degrees higher. Thus I could see the Sun and an apparently full Moon in the sky together.
Bu there's more!
One time I was in a low place, surrounded by hills, buildings, and trees, with much less than a full hemisphere of the sky visible, about 9 or 10 in the morning. And I saw the moon, and it looked round and full as far as I could tell. And of course the Sun was also visible. And it looked like the Sun and the Moon were much less than 180 degrees apart, despite the Moon looking full as far as I could tell. And if the Moon was close enough to full to look full to me, its apparent magnitude was probably close to -12.50.
So the problem is to increase the light scattering of the atmosphere of your planet enough increase the brightness of the sky so that the apparent magnitude of the full moon of your planet is reduced by about 8 magnitudes when the Sun is high and reduced by about 10 magnitudes when the Sun is low in the sky.
So you may need an atmospheric expert to calculate how much water vapor, dust, various gases, etc. you may need to add to your planet's atmosphere to make your planet's full moon invisible in the day sky, and whether that atmosphere will still be breathable.
And you can make your planet's moon a bit dimmer, which would help a bit. Since we don't know for certain how important a large moon is for planetary habitability you might decide to make the planet's moon much smaller than Earth's Moon, or might be afraid to make it too much smaller than Earth's Moon.
But you should be able to reduce the mass of the planet's moon to be considerably less than that of Earth's moon. Then increase the average density of the planet's moon so that the same mass can be within a smaller volume. Those two effects should make the planet's moon significantly smaller than Earth's Moon and thus it would look much smaller and be dimmer at the same distance as Earth's Moon.
And you can lower the planet's moon's albedo, the amount of light it reflects, to make it dimmer. Unfortunately, Earth's Moon already has a low average albedo of 0.137. Dark solar system objects like comets and asteroids have albedos down to 0.05 and 0.05. This can significantly reduce the apparent brightness of your planet's moon.
When the Moon is new and near the Sun in the sky, the Moon's visible side is almost totally in shadow. But sunlight reflected from the Earth, and thus becoming Earth light, does light up the Moon a little and make it a little brighter when it is new. The albedo of the Earth is about 0.30 to 0.35.
Decreasing the albedo of your planet would decrease the amount of light the planet reflects back onto it's moon when the moon is new, and thus will make the moon somewhat darker when new. It would also mean the planet could be slightly farther from its sun and have the same temperature as the Earth. Thus there would be less sunlight for the planet's moon to reflect back to the planet, making the moon dimmer.
I am not sure how well decreasing the planet's albedo would fit in with making the atmosphere scatter more like and obscure the light of the moon during daylight.
So changes in the atmosphere and in the planet's moon might make the planet's moon visible in the night sky and invisible in the day sky and even at sunset.
So making the planet's moon less bright and making the planet's atmosphere obscure the moon more might be enough to make the planet's moon visible during the night and invisible during the day.