Ah interesting, thanks - I did notice that little spot you've captured out of the video on the first day that I got her, but concluded (wrongly it seems) that it didn't look like shell rot to me and must therefore be some kind of birth mark. I've just had a close look at it again, and that spot looks exactly the same today as in that video which is nearly three months old, so that's probably a good sign.Hair algae hopefully would out compete it but if its growing there's def nitrates feeding it.
Lighting is more or less just a correlation, your setup is fine but most keep them with inproper burrows so they're more 'exposed and stressed' without the dark areas to feel secure in. I don't think keeping one in a reef tank vs. no lights are all would be any different if the water quality is perfect. Lighting implies shell rot is photosynthetic which is possible but we don't really know for sure. Big mantis shrimp are messy so nitrate spikes are a given and shell rot development can be as small as a tiny spec that either grows or scars.
Get some macro algae that's pretty to look at, like dragon's breath, mangrove, etc, mantis shouldn't bother them and they'll drain nutrients out of the water for hair algae, starving it, would substitute for a CUC that way.
EDIT: I just watched one of those videos to see if I could spot anything, this is the spot you need to watch for, this is def shell rot on the carapace above the right arm, check if its still there after the molt or if it scar'd:
about 0:07 you can see it in the 26second video.
Good idea with the macro algae - I've heard that suggested before for managing nitrates but didn't make the connection with out-competing the less desirable algae in the tank. I'll look at some different macro algae options - I don't think my LFS sells it for some reason which is the main reason I haven't taken the idea any further yet, but I'm sure it'll be easy enough to find some online. I presume it'll be fine to stick it towards the back of the tank and keep the rockwork free for corals?