Hi Erevos,
Please take my evaluation with a pinch of salt as I'm only playing these stakes too! I personally think that the range you give is way too pessimistic. Villain has a VPIP of 44% so he's got lots of things in his range preflop (although I think PFR and 3bet stats would also be useful to determine how aggressive he is and whether he is likely to 3bet hands like JJ).
In my experience so far, a flop min raise is nowhere near as nutted as a turn min raise. Sometimes players do it to buy themselves a cheap card when they're drawing, or to take away your initiative when they have marginal holdings. Of course there are some monsters in his range here, but I think he can have plenty of pair+draws combos such as QT, QJ, KJ, KT. There are also some two-pairs in his range, against which we've still got some limited equity. I would also add in AJ.
My range would be something like:
Board: ThJs9c
Equity Win Tie
UTG+1 44.63% 43.64% 1.00% { AhAs }
SB 55.37% 54.37% 1.00% { JJ-88, AQs-AJs, KTs+, Q9s+, J9s+, T9s, 98s, 87s, AQo-AJo, A8o, KJo+, QTo+, JTo }
The problem, in my opinion, is that by calling you give yourself a bit of a problem as it's going to be so hard to play the turn. If he checks to us, it's hard to know whether to bet again and possibly face another raise or to check behind and give a free card away on a drawy board. If he bets then it's hard to call knowing that we're probably going to face another bet on the river.
That said, as played, I really don't think you can call the flop and then fold to such a small bet on a blank card. But I think I'd prefer 3bet/folding the flop, making it around $1.05 as I think that we can get value from his pair+draw type hands (my plan would then be to bet just over 1/2 pot on a blank turn for a free showdown). If he 4bets us then I'm happy to let the hand go.
Found it an interesting spot. Please feel free to flame if I'm profoundly wrong!