I think the range you assign him preflop is fine. He might even flat you wider, so you could include some weaker AXs, SC's and PP's that he might call IP for implied odds. I think the tightest he would 3b would be QQ+ and AKo+ in this spot so I would not include these in his flatting range. I would expect him to have called preflop with something like:
JJ-22,AQs-A2s,K9s+,Q9s+,J9s+,T9s,98s,87s,76s,AQo-A8o,KTo+,QTo+,JTo.
Worst case scenario for his flop calling range IMO is that he plays fit/fold OTF and only calls w/ overpairs or better in which case we are boned:
Spoiler
Board: 9♥9♦5♣
Equity Win Tie
MP3 14.69% 13.53% 1.16% { 7h7s }
CO 85.31% 84.16% 1.16% { JJ-TT, 88, 9s9c, 7d7c, 5d5h, 5d5s, 5h5s, As9s, Ac9c, Ks9s, Kc9c, Qs9s, Qc9c, Js9s, Jc9c, Ts9s, Tc9c, 9s8s, 9c8c, Ad5d, Ah5h, As5s, Ad9s, Ad9c, Ah9s, Ah9c, As9c, Ac9s }
More likely is that he calls with all sorts of floats and any pair. Something like over cards with good backdoors, gutshots, decent A-hi that could still be the best hand from his point of view and any pair+:
Spoiler
Board: 9♥9♦5♣
Equity Win Tie
MP3 61.38% 60.76% 0.62% { 7h7s }
CO 38.62% 38.00% 0.62% { JJ-22, T9s, 98s, 87s, 76s, 65s, AdQd, AhQh, AcQc, KdQd, KhQh, KcQc, AdJd, AhJh, AcJc, KdJd, KhJh, KcJc, QdJd, QhJh, QcJc, AdTd, AhTh, AcTc, KdTd, KhTh, KcTc, QdTd, QhTh, QcTc, JdTd, JhTh, JcTc, Td8d, Th8h, Tc8c, AQo-AJo }
As for whether to barrel the turn or not I think we need to determine whether or not we turn our hand into a bluff by barrelling and if we need to. Personally I think our hand is good enough to x/call here. I think barrelling would unnecessarily turn our hand into a bluff. I would not expect villain to call a Turn barrell here with worse. I think an argument can be made for that fact that we may fold out villain's equity share vs hands that don't have us beat but have a chance to OTR and also betting ourselves prevents us from being bluffed here or on future streets.
However, if we check ( although villain is a relative unknown so we don't know for certain) we give villain the opportunity to bluff at a scare card. Villain could be the type to float us a lot in this spot ( I think it would be a very good spot to do so). As a result I would be inclined to x/c if we think he is capable for bluffing here and re-evaluate OTR expecting villain to x/beh OTR un-improved. Or just x/fold if you think he rarely bluffs here. It's kind of opponent dependant IMO, but x/c would be my standard line here.