Perhaps I'm being too nitty but I would c/f OTF and make a smaller mistake than a potentially bigger mistake. I think that double barrelling, even just single bet on the OTF would be a bigger mistake in the long term; here's why:
1) I agree that OTF the opponent could have 77-22 but I think the majority of his calling range would be 10x, 9x, random broadways SD's. If we're unimproved, we're losing to trips, made pairs or K/Q high. Even if we are improved, we could be losing to higher straight or a slowplayed full house.
2) We only have 8 hands on our opponent so we don't know his player type yet, but even if he's a calling station calling down w/ 77-22, would we make a straight by the river enough times to show a profit in the long term?
3) If we assume he's more of a thinking player, would he really call one street w/ 77-22? If he's a thinking player, he would know that 77-22 are very susceptible to overcards and by peeling one street he would expect the potential for more overcomes to come in the next two streets. It would seem silly to call OTF and not expect more overcards to come later. So I would imagine a player like this would fold 77-22 OTF.
We have an OESD on a board that doesn't hit that many ranges given that it's paired. C/f'ing the flop is way too tight. Betting is certainly +EV.
We also shouldn't fall into the process of "waiting for betting spots" which it kind of sounds like you're going down that road here. Sorry if I'm off with that assumption. That logic really applies to tournaments, but not cash games. In a tournament you can be in situation where the EV is so thin that you should not make the play because the EV of staying alive is higher.
The more I think about it, the more I'm ok with c/f'ing the turn on this particular card, but definitely not just c/f'ing the flop. I think it's really close on the turn. I'll flesh out my thoughts in response to your points.
1) I think there's a decent amount of 77-22 just from a combinatoric standpoint. There's 39 combos of 77-22 if he always call the flop with them. There's not that many Tx. AT, KT, QT, JT, T9s, T8s = 34 combos. 9x is A9, K9s, Q9s, J9s, 98s, 97s = 27 combos. Thats 39 combos of 77-22 and 61 combos of Tx and 9x. So if we expect him to fold 77-22 100% and never fold 9x and Tx then we'd get (39/(39+61)) 39% FE. The problem with this turn card is that he also has some QJ that's never folding, and he could also have hit with KQ is he is calling wit that. That puts us pretty close, and into the 35% FE range. That makes it really close on a 2-barrel. Sometimes we raised on the turn and sometimes we hit the wrong end of this straight and lose another bet. I would say we get right around 25% FE on the turn... so it's pretty close. (Double-check my combo counting... a lot of opportunites to mess up)
2) I'm wouldn't expect to get called twice here vs those hands, but if he does have those hands then we'll likely showdown unless we make a straight, and in which case we win. We have a lot of outs vs those hands so it's not the end of the world if they call - thought we obviously prefer they fold. We can put in a fudge factor in the above analysis to account for this. So, instead of always folding 77-22 let's say he folds them 80%. Instead of 39 combos of those hands folding, we now have 31.2 (39*80%). Now he folds (31.2/(31.2+61)) 34% FE. Plus he has some QJ that's not folding, so maybe more like 30% FE. He also has some random hands like 76s and 86s and AK that might consider folding the turn. I think it kind of washes out a little bit in decreasing our FE a few points. If we 2/3's it on the turn that isn't quit enough on pure air, but we do have some outs that reduces our FE required. Pretty close! There's also something to be said for checking the turn is definitely +EV because we do get a free card from time to time and then hit a straight. Seems like a check to me now.
FWIW, if we do end up showing down this hand there's some EV attached to that because it gives us information for relatively cheap that we can use in the future.
3) Giving a player at 25NL a lot of credit for thinking is likely a mistake. On average most guys are pretty ABC or they are fishy. This player looks fishy. Given this guys stack size I expect him to be a pretty weak player, so I wouldn't go very far along that train of thought.