Doesn't your range look a ton like {broadways, AA-JJ} on the flop? It would make sense that you check most of your air on the turn here...doesn't that make a c/f bad since he would actually bet his JJ-TT for protection?

Also, you say that b/f'ing would be bad unless we b/f a small amount. It seems we can get value from JJ-TT by betting a small amount since he probably doesn't shove these hands much (a shove is probably something more like {99, KK+} which we're pwnd by. I don't think a b/f is as bad as you make it seem.

What about a c/c turn, c/f river line (pending on bet-size)? Again, I believe he'd be protecting the turn here a lot and checking through the river with hands we're looking to get value from. He probably would see a lot of showdowns with these hands meaning he wouldn't bet them on the river much (if he bets river, we're probably beat).

If the turn checks through, betting the river a smallish amount ($25-$28?) would garner a lot of calls from JJ-TT.