I find that the best thing to photograph during a full moon is either really bright nebula where the moon has no real effect. Either that or shoot star clusters.
To give an example, this
Eta Carina image was shot under full moon from my bortle 7 backyard. It's bright enough that a lot of the nebulosity isn't hugely affected. Yes, it would have stronger signal and less noise under dark moonless skies but you take what you're given sometimes.
As another example,
NGC 253 was not only shot under a near full moon but the moon was only some 20º away if memory serves; this kinda makes the light polluted backyard meaningless. I did this mostly because I was curious about how close I could get to the moon and still achieve a result.
Star clusters are the best to chase though as light pollution and the moon don't do anything other than potentially wreck some havoc with colour calibration as you're not worrying about the background signal.