So, let's first compare the distances. Your average LEO satellite is placed in orbit maybe 600 km. The distance to the moon is 384000 km. That is 640 times further. Using $r^2$ losses, and converting to decibels, that is an additional 56 dB loss. Of course, a groundstation needs to be able to communicate in LEO beyond directly overhead, so let's just assume it's at the horizon at that distance. I'm going to estimate that horizon distance at about 3000 km. That reduces the difference in power requirements to just 42 dB difference. So, is that achievable for an amateur station?
Most cubesats are build by students or other hobbyists, and use the Amateur Radio frequencies to communicate. There is actually a pretty common feat to test out an amateur radio station that involves pointing an antenna at the moon, and trying to communicate with other people. For the most part, the data rates are quite low, but a large station can actually talk using morse code, which can be fairly fast. This is actually quite a bit harder than talking to a satellite orbiting the moon, for a couple of reasons. First of all, the moon don't reflect all of the energy back to earth, and then you have to consider the two way path loss in signals. So, it can in theory be done, so, how do you do it?
First of all, you will need to use a lower data rate than the LEO satellite. If you cut the rate down by 16, that's about a 12 dB gain. If you use a preamplifier/amplifier, you can get another 20 dB or so. And then using higher gain antennas will get you the rest of the way, all the way to the required 42 dB extra gain. The lower data rates could be compensated by the fact that you have near constant communication for 12 hours each day. Alternatively, you could use a higher frequency, and a dish antenna, which will allow you to get much higher gain, and a similar feat could be ensured.
Bottom line is, this is a doable problem, although it would be a bit tricky to manage. It should be tested first by bouncing a signal off of the moon and verifying you hear the echo. This should be done using the same modulation you plan on listening with. If you pull this off, you should be good to go.