So a bit of an update. Since I was no longer using my RPi3 (I had ordered a XU4) I decided to experiment with it (the RPi3) more than I had before and I installed Lakka on it, and it worked perfectly. Out of the box the games look amazing and the (few) that I have tried, both SNES and PS1 have run perfectly.
This tells me that there isn't a problem with the pi, the power supply or the sd card, its just something different in the versions/configuration in RetroPie. For clarity sake, I always installed RetroPie from official builds, and tried multiple times, always trying to use it with out of the box defaults first.
I gave the RPi3 away, so I can no longer test with it (the XU4 runs circles around it) but it is good to know that it was at least capable of running the emulators that I thought it wasn't.
I have not put RetroPie on the XU4 yet, I am trying to get a decent set of benchmarks (games that run smoothly in Lakka) to compare.