As is well known, and as you can see in the image below, the far side of the Moon is brighter than its near side. What is its overall albedo? And what would be the magnitude of a full moon if the (current) far side were facing us?
This is a bit rough, since I can't find the exact numbers anywhere. However the lunar highlands have an albedo that is double the lunar mare (about 0.2 compared to 0.1)(source). The far side is almost completely highland, so its albedo is about 0.2
But only about 1/3 of the nearside is lunar mare (source).
So the relevant calculation is $(1/3×0.1 +2/3×0.2)\div 0.2 =0.83$ That is the nearside is 83% of the far side in albedo. Or the far side is 20% brighter.
In terms of magnitude, a 20% increase in brightness corresponds to about 0.2 magnitudes brighter. The full moon has magnitude -12.7, so a far side full moon would have magnitude -12.9.
These are fairly rough figures, but the impression is that a "far-side" moon would be brighter, but not exceptionally brighter than the moon as we see it now.
We may be able to shed some light on this question by looking at the magnitude of the Moon from the Lunar Reconnaissance Orbiter (LRO).
Ideally, we'd like data from a satellite in a high altitude equatorial orbit, but the LRO is in a polar orbit (inclination ~85.2° to the lunar equator), and it has a rather low altitude. Its semi-major axis is ~1830 km, but the Moon's radius is ~1737.4 km. Wikipedia claims that the LRO is in an eccentric orbit, but that information is out of date, and its current eccentricity is ~0.00129, but admittedly its altitude deviates from that of a simple Kepler ellipse, due to the lunar mascons. The inclination of its orbit is currently quite stable, though.
The LRO has JPL Horizons ID -85. Its trajectory data was recently updated, on 2023-Aug-3. You can see its osculating orbit elements for 2023-Aug-1 18:30 TDB here. That time is close to the most recent full moon (2023-Aug-1 18:31:40 UTC).
The LRO has a sidereal period of almost 2 hours (1h 57m 2.9s). Here's a plot showing its distance from the centre of the Moon, covering 6 hours, with a 2 minute timestep.
When it's full moon, the far side is dark. When it's new moon the far side is fully illuminated. So we can get a rough idea of the difference in albedo by looking at the magnitude of the Moon at those times.
Here's a Moon magnitude plot covering the same timespan (using the same timestep).
And here is the corresponding plot for the previous new moon (2023-Jul-17 18:32:50 UTC).
The mean magnitude on the new moon is -21.466, which is slightly brighter than the full moon mean of -21.168. Interestingly, the peak magnitude at full moon is slightly higher than at new moon. The minima during the new moon are noticeably brighter that at full moon. I assume that's mostly due to earthshine.
However, Horizons doesn't say how it computes these magnitudes, and it gives this warning:
Moon's approximate apparent visual magnitude and surface brightness. When phase angle < 7 deg (within ~1 day of full Moon), computed magnitude tends to be about 0.12 too small.