The pixel size of the HST's Wide Field Camera 3 or WFC-3 is 164/2048 = 0.08 arcsec.
The night side of the Moon is illuminated by Earthshine and the brightness depends on the phase angle of the moon (and the weather on Earth (clouds, wind-induced waves on the ocean) and the time (ocean versus land) but we can find some averages. Let's use +15 magnitude per square arcsecond from below with a as a representative number. It gets a bit brighter at small phase angles when the Earth's mostly sunlit side faces the Moon, but then the Hubble would be starting to point dangerously close to the Sun.
A pixel size of 0.08 x 0.08 arcsec is 0.0064 of a square arcsecond, so that's $-2.5log_{10}(0.0064) \approx +5.5$ magnitude; so its +20.5 magnitude per pixel.
The Sun is +27 magnitude, 1360 W/m^2, and 2.5E+06 square arcseconds. Assuming HST+WFC-3 are used without a filter, they cover most of the power in the solar spectrum.
So the Sun's -11 mag/arcsec^2 or -5.5 mag/pixel is 5E-04 W/arcsec^2 or 3E-06 W/pixel.
That makes the night side of the Moon (26 magnitudes dimmer) about 1.4E-16 Watts/pixel.
Multiply by 6E+18 (e- per Coulomb) and divide by 2.5 eV (some kind of average energy per photon) and that's about 200 e- per second in the CCD if I throw in a quantum efficiency of 0.8 and a pixel fill-factor of 0.8.
That's fairly dim! A five second exposure gives 1000 e- and the $\sqrt{N}$ shot noise is 3%. The cooled CCD's dark current is minuscule here, we're limited by shot noise: 1, 2 and readout noise of roughly ~20 e-.
As the other answers have pointed out, tracking might be a challenge. Kinematically there's no difference between the telescope slewing at 0 arcsec/sec to follow the stars and 15 arcsec/sec to follow the Moon, but there may be problems with the way the star cameras obtain data if the stars are moving, and so it may be necessary to give the HST a "blind kick" of 15 arcsec/sec in order to follow the Moon, and read out images say once per second to off-line stack them correcting for any residual sub-pixel drift.
From Measurements of the Surface Brightness of the Earthshine with Applications to Calibrate Lunar Flashes (also here):
Abstract
We have used the large database of photometric observations of the bright and dark portions of the face of the Moon from the Earthshine Project at Big Bear Solar Observatory to determine the surface brightness of the earthshine and its variations. Our purpose is to make these observations appropriate for the calibration of lunar flashes according to their magnitude. We have evaluated the daily, seasonal, and annual changes in magnitude for our entire data set and have also calibrated the surface brightness of the entire lunar geography for several lunar phases by means of the observation of lunar eclipses. We find variations between +12 and +17 mV arcsec² with hourly changes upward of the order 0.25 mV arcsec², which are uniquely due to the terrestrial meteorology. This rapid change in the terrestrial flux reaching the Moon is usually neglected when calibrating the magnitude of lunar impact events. We justify this using earthshine observations to determine the brightness for the day, time, and selenographic location of a given event in order to improve the accuracy of its brightness calibration up to 0.25 mag.
