Identifying an eccentric bottom bracket?
Pulling an eccentric out that's not stuck is usually pretty straightforward once you've gotten the cranks off. If you want to find out the brand and model info (which might not actually be that useful anyway), that's probably the only place you'll find it, if it's there. But, if you're going to try to source a new eccentric yourself, you really should pull it anyway to get an exact measurement of the shell's inside diameter.

If this is any kind of mainstream bike manufacturer, where I would start here is calling them up and seeing if they have a replacement they can send to either you or a dealer. If so, that path is going to be about a million times easier than anything else, and probably cheaper.

I'll also mention that the framebuilding suppliers like Nova are the main source I've seen for more simple 1-piece eccentrics like you appear to have. Most of what you'll find elsewhere are the premium ones like Bushnell, Phil, and Niner.

As for the Loctite idea, with GXP there's not really any place for it to move to. I would never recommend riding on a loose cup, but people do all the time, with the main consequence being noise and play in the cranks.