For an SF novel, if there were two large moons orbiting a planet, let's say one moon the size of Earth's moon and the 2nd moon about 20% larger, and the planet roughly the size of the Earth, would the L2 point at the larger moon still exist and be relatively stable as it is in the Earth-Moon system (at least in the short term, e.g. for a matter of days)? You can posit the moons as orbiting opposite to each other around the planet, or you can posit them in resonance, I don't care as long as they are not too close to each other.
The Earth-Moon Lagrange points are defined (I didn't use the word "exist") based on the gravity of the Earth and the Moon. If you forget all the images you've seen drawn in rotating frames and just think about the real world, all five of them are really just orbits around the Earth with a period equal to the Moon's period. They are slightly nudged and prodded by the Moon's gravity so that they stay in 1:1 resonance with the Moon, but they are truly Earth orbits.
That is also true for "halo orbits", which look like they orbit the empty space where L1 and L2 are supposed to be, but in a non-rotating frame they are also just orbits around the Earth that weave up and down slightly due to the Moon's nudging and prodding.
While there are in fact some halo orbits that are stable in a pure, mathematical circular restricted three body problem (Earth-Moon orbit is purely circular, nothing else exists in the universe), in the real world the Sun's gravity, and the elliptical shape of the lunar orbit mess with this, as do several other effects. So spacecraft in halo orbits will require regular station-keeping maneuvers to stay there.
- Are (some) Halo Orbits actually Stable?
- Halo orbits are just the beginning of the kinds of orbits possible in three-body systems! See this answer and this and this for a peek into the bowl of spaghetti that is the possible three-body orbits that a clever author with dreams of a snazzy screenplay deal could choose from!
- If the two moons are in orbital resonance, then there is also the mind-blowing possibility of four-body orbits that are closed (i.e. they repeat) but that's beyond the scope of this answer.
The stability of halo orbits around the mathematical Lagrange points associated with one Moon will be strongly degraded by the presence of the other Moon of similar mass. How bad the degradation is depends both on the difference in distances from the Earth (farther the better) and on the ratio of the periods, and the relative directions!
If the period ratio is a rational number (e.g. 2:1, 3:2...) then there may be unusually strong or unusually weak destabilization compared to other ratios, and this would also depend on the period of the halo orbit (usually roughly a half of the Moon's period but there is a wide range, depending on the size of the halo).
See for example this answer to a different question, which suggests that if the perturbing moon were retrograde (orbiting in the opposite direction of the Moon with the halo orbit) the destabilization might be much less!
However, a detailed answer to your question really does require a deep dive into the technical literature and/or direct numerical simulations. Hopefully orbital mechanics will leave additional answers here!
As an aside, you may find this answer to the question Do we sufficiently understand mechanics of Lagrange point stationkeeping for EML2 rendezvous and assembly? to be interesting as well.
And now for something completely different:
- What is a near rectilinear halo orbit?
- Why is a near rectilinear halo orbit proposed for LOP-G (formerly known as Deep Space Gateway?)
- How will the Lunar Gateway go to L2 and L1?
- How to find Near Rectilinear Halo Orbits (NHROs)?
This is one of my all-time YouTube favorite videos. It's an illustration of the Earth-Moon Lagrange points in orbit around the sun for a year. Put on your headphones, dim the lights, and set YT to HD.
- See answer(s) to What do the green lines represent in this Lagrange Point animation? for more about the video itself.
Below is from this answer to the very interesting question What determines the orbital speed around a massless Lagrangian point?