I'm coming around to the fact I'll probably have to use CA too. My Epi LP has a couple of lifting inlays.
Having read through countless threads on how to reglue inlays the choices seem to be CA, epoxy or PVA (white wood glue).
I tried Elmer's wood glue and clamping but the inlays didn't remain stuck down. Right now I have electrical tape over the raised inlays so I don't snag them with a string or finger when playing.
I've found it difficult to reach for the superglue as the manufacturer didn't appear to require it originally, but I think my reticence could be OCD at play.
Just my thoughts on a similar issue.