Whatever you do, if you think betting the river would be a bluff then ofmg do not bet! I think this can easily be JJ+ as you have suggested, a cbet of 55% could be indicative of him bluffcatching a lot with surprisingly strong hands.

It really depends on the history between you, if he thinks you are capable of floating/ firing river w/ air then he's more likely to be doing this with 88/99 intending to call. If you have a nitty image then simply check behind, there's no shame in checking back an overpair.