Spoon, don't get me wrong, I am not pre-occupied with being right, I am pre-occupied with getting it right. I am perfectly willing to accept that I am wrong, but not until I perfectly understand why.

So, I will do this range exercise you propose, and in the process I am going to try to show (as mathematically as I can) why I am not comfortable with the check play.

We will make the following assumptions:
- I start with the preflop range of the OP, as I accept it is reasonable
- if we bet the flop, no better hand folds and no worse hand calls
- if we do not bet the flop, we will call (and not raise) bets by our opponent as dictated by pot odds and our equity against his estimated range. We will do so on both the turn and river.

Case 1: We check back the flop. We estimate that opp's betting range on the turn is 44,88,99-TT,AXs,98s,87s (as per OP) and we are right (I mean that the range is correct). This implies that we think opp never bluffs on the turn. As pointed out, this is 66% of his original range and against this range we have 11.3% equity. We assume opp always bets with this part of his range and is not stupid enough to bet so small as to give us calling odds. Against his no betting range (44% of original range), we have 83% equity. For simplicity, we will assume for this part of the range that at no point both opponents will put more money in the pot, so the pot size stays $1.20 until the end and 83% of the time we will win it and 17% of the time opp will win it. What is our total EV here:
EV = 0.66*0 + 0.44 * (0.83*$1.20 + 0.17*0) = $0.4382

Case 2: We check back the flop. We estimate that opp's betting range on the turn is his entire preflop calling range and we are right (the range is correct). This implies that we think that opp always bluffs his air on the turn. Against his entire range on the turn we have 71.6% equity. We will also assume that he continues to bluff on the river even if we call the turn, whatever the river card is, and let's be optimistic and assume he will bluff his entire stack, and he will also bet his entire stack when he does not bluff. We call all the way. No raising. Our EV here is:
EV = 0.716 * ($1.20 + $3.75) + 0.284 * (-$3.75) = $2.4792

Case 3: We bet half pot on the flop. He continues only with 88,44,ATs-A2s,AJo-ATo, which is about 32% of his starting range. So he folds 68% of the time and, worst case scenario, he raises the other 32% and we fold.
EV = 0.68 * $1.20 + 0.32 * (-$0.6) = $0.624

Case 4: We check back the flop. We estimate that opp's betting range on the turn is his entire preflop calling range but we are wrong. His turn betting range really is that of case 1. 44% of the time he will not bet and we will know we were wrong and correct the range to the actual one. 66% of the time, he will bet and we will think we have 71.6% equity against his range when we really have only 11.3% equity. So we will wrongly call all his bets to showdown.
EV = 0.44 * (0.83*$1.20 + 0.17*0)
+ 0.66 * (0.112*($1.20+$3.75) + 0.888*(-$3.75)) = -$1.3937

Case 5: We bet half pot on the flop. He folds 68% of the time (all hands we beat) and calls 32% of the time (all hands that beat us). The turn always goes check/check and he checks the river. By the river, we hit a set 8% of the time and extract an extra half pot bet half of the time and get nothing more the other half of the time. 46% of the time we bluff the river (half pot bet) and he folds. The remaining 46% of the time we bluff the river for half pot and he calls and let's say we always loose for simplicity and conservatism.
EV = 0.68*$1.2
+ 0.32 *(0.08*(0.5*($2.4+$1.2)
+0.5*$2.4)
+0.46*$2.4
+0.46*(-$1.2) = $1.0694


Now let's combine case 1, 2 and 4:
- If we always assign him the tight range, our EV is $0.4382.
- If we always assign him the wide range and we are wrong half of the time and right half of the time, our EV is
EV = 0.5*$2.4792 + 0.5*(-$1.3937) = $0.5427
- If 50% of the time, we assign him the tight range and the other 50% of the time, we assign him the wide range, and we are right 50% of the time, but wrong the other 50% of the time. What's our EV?
EV = 0.5*$0.4382 + 0.5*$0.5427 = $0.4904

So apparently, if in doubt, it is better to always assign him the wide range and call his bets all the way. In play, I would not f*cking know which range to put him on because I would generally not have good enough reads to estimate a correct range. I defy you to ask 10 people for a turn betting range in this spot and get less than 10 different answers.

For case 3 and 5 let's say that case 5 never happens. We are always raised by better hands on the flop and fold. Our EV is still more than that of any of the checked flop scenarios.

So in the end I can't help finding more EV in betting this flop. I just spent two hours doing this instead of playing and I honestly wish I could have proved myself wrong here to end this conundrum but I don't seem to be able to. I think I have made enough efforts to deserve a clue to put me on the right track. Where do I go wrong here?