TLDR: The primary ~28 day lunar orbital librations are well modeled by taking into account the changing orbital speed due to the Moon's orbital eccentricity. The forced physical librations (which include Cassini's 2nd and 3rd laws) can be modeled closed form with assumptions about the Moon's interior layers. The free librations can be fitted to Lunar Laser Ranging data.
Setup: Even though the Moon's rotation rate is the same as its orbital period, its orbital speed changes due to its orbital eccentricity. It moves faster at perigee than apogee in accordance with Kepler's 2nd law. This miss-match between orbital speed and rotation rate is the primary contributor to lunar libration and (as expected) cycles with the Moon's orbital period. This libration isn't quite along a single axis due to the 6.8 degree obliquity of the Moon's rotation.
The Moon is tidally deformed by the Earth. This tidal deformation axis leads or trails the vector from the Moon to the Earth through Apogee and Perigee, respectively, leading to torque on the Moon.
Earth's gravitational field is irregular. The Earth is well modeled by an oblate spheroid and subject to its own tidal deformations mainly due to the Moon and the Sun. The orientation of the Earth's lunar tidal axis to the Earth/Moon vector is mostly transferring angular momentum from the Earth's spin into the Moon's orbit. However, it is also imparting torque to the Moon (an irregular shape in an irregular gravitational field).
Gravitation from other bodies in the Solar System, even when treated as point masses will cause tidal torques on the Moon. I did a little alteration to Bate, Mueller, White's gravitation table to show gravitational acceleration from some of these other bodies on the Moon.
Note the tidal torques are a function of both gravitational force and gradient, so even though the Sun exerts higher gravitational forces on the Moon than the Earth exerts on the Moon, the Earth exerts more tidal torque since its gravitational gradient is higher. The sum of all these torques affect the rotation of the Moon and thus librations.
Also, we can't separate the long term lunar orbital effects from libration, since eccentricity is a primary cause of libration. Normally in a two body system we would expect circularization of the lunar orbit. But for our Moon, the eccentricity is actually growing due to gravitational interactions from the Sun!
Answer:
The librations of the Moon are the result of the changing geometry of the Earth/Moon system due to the sum of forces on the Moon combined with its orbital and angular momentum. The Moon is irregular in shape and composition, and changes shape due to tidal forces. It also passes through an irregular gravitational field since the Earth is similarly deformed. This causes various torques on the Moon, which result in changes to its rotation.
However, the dominant force of tidal torque prevents the average orbital and rotation periods from ever deviating. Hence, the librations.
Pavlov et al. 2018 wrote a paper called Determining parameters of Moon's orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models. Here are the parameters they use:
I am surprised they don't explicitly include "solar radiation pressure" explicitly as one of the forces on the Moon, but perhaps it is hidden in one of the other variables. Instead of using a complete physical model for the free physical librations, they use various models to fit to the LLR (Lunar Laser Ranging) measurements.
For the forced librations (including Cassini's 2nd and 3rd laws), there are some self-consistent formalisms, like in https://arxiv.org/abs/2201.00803. However, they require some assumptions about the mantle and core of the Moon and they are extremely complex.
