The deviation in the Moon's position due to the reduced light pressure during a lunar eclipse is too small to measure, even with Lunar Laser Ranging (LLR). However, the reduced surface temperature may have an effect on the lunar orientation (hot rock expands, raising its centre of gravity), but I suspect that would be hard to separate from the effects caused by the increased tidal force on the Moon when the Sun, Earth, and Moon are aligned during an eclipse.
The JPL ephemeris calculations use a very sophisticated model of the Moon. The ephemeris is calculated by integrating the equations of motion of the major masses of the Solar System (including 343 asteroids), and then fitting the numerically integrated orbits to ground-based and space-based observations. For details, please see The JPL Planetary and Lunar Ephemerides DE440 and DE441, Park et al (2021), DOI 10.3847/1538-3881/abd414
in particular, section 2.4 Orientation of the Moon, and section 4 Rotational Dynamics of the Moon.
The motion of the Moon is quite complex, and we cannot model it analytically to the precision matching the LLR measurements. As I said recently on Astronomy.SE, the lunar theory of E. W. Brown, developed around the turn of the twentieth century, used over 1400 terms. The more recent Éphéméride Lunaire Parisienne (ELP), developed by Jean Chapront, Michelle Chapront-Touzé, et al, "contains more than 20,000 periodic terms, [but] it is not sufficiently accurate to predict the Moon's position to the centimeter accuracy with which that can be measured by LLR". Note that ELP was not fitted directly to observations but to the JPL DE, initially DE200, but it was later updated to DE405.
It's kind of hard to spot anomalies in the Moon's motion because our lunar motion models have been adjusted to match our observations. I suppose there might be terms in the ELP equations that correspond to light pressure, but I haven't studied the details of that theory.
Light pressure is important for modelling the motion of small bodies, including spacecraft, but its effects on large bodies is negligible, and effectively causes a slight reduction in the body's attraction to the Sun.
According to Wikipedia's article on radiation pressure the solar radiation pressure at 1 AU is ~$9.08×10^{-6}\,\rm N/m^2$. We can estimate the force this has on the Moon by multiplying that by half the Moon's surface area (~$3.793×10^{13}\,\rm m^2$), which gives a force of ~$1.722×10^{8}\,\rm N$. Dividing by the lunar mass (~$7.342×10^{22}\,\rm kg$) yields an acceleration of ~$2.345 ×10^{-15}\,\rm m/s^2$, which is pretty tiny. Over a 2 hour lunar eclipse, such an acceleration would cause a displacement of just under $61$ nanometres. (The lunar parameters in the above calculations are from Wikipedia).
I'm being generous in that calculation. We probably should use the cross-section area, which is approximately one quarter of the surface area. On the other hand, the pressure is doubled for a perfect reflector, but I've ignored that because the Moon's albedo is only ~$0.136$, so only $13.6$% of the sunlight is reflected and the other $86.4$% is absorbed. Also, at Full Moon, the Moon is further from the Sun than the Earth is, but the Moon's orbital radius is smaller than the variation in the Earth-Sun distance, and the radiation pressure value we're using is derived from a mean value for the solar energy flux.
I should mention that it is difficult to make LLR measurements at Full Moon. The LLR detectors only receive a small number of reflected laser photons, and at Full Moon the signal is swamped by sunlight. Also, the two Lunokhod reflectors are unusable at Full Moon, due to thermal expansion, so only the three Apollo reflectors can be used. However, the LLR observing stations do try to take advantage of lunar eclipses.
I should also mention that although we know the distances of the lunar laser reflectors to sub-centimetre precision, we do not know the location of the Moon's centre of mass to the same precision. At the centimetre scale, the Moon is not a rigid body, it's a big wobbly ball of looney mooney... stuff. ;) As mentioned in Park et al, the JPL model does account for the Moon's liquid core, but it's difficult to do that accurately over timespans >1000 years.
