I don't see why he can't show up here with 77 or 44, but even adding that, I think I like check/fold. I don't think villain is showing up with air in this spot, so he'd have to then be the type of player to actually acknowledge his modicum of showdown value is no good and to bluff with it. I don't know how often you see that in a live circuit tourney. I think he's more likely to be check happy with the range you described. If he won't bluff, you get to play tourney life control on this bad river.

If his range is (JQ/KQ/AQ, 88, 99-JJ, 77, 44, QQ), and you think he's calling off all Queens, then he'll have 36 combos of worse hands calling, and 69 hands total. 52% of hands you betting into are calling and losing to you, so it's profitable.

If however, his range loses any queens, it falls to below 50% of his range and becomes unprofitable.

I know you have a 2/3 pot sized bet left, but what about betting something very small and folding to a shove. It may get calls from his 99-JJ if you bet something like 1200 into 6000. This really hinges on him being stationy/rec playery though. But for that sizing, it can increase the hands you beat that call by a significant % of his range, and if he very infrequently bluffs, you can fold. He'd have to not bluff more than 25% of the time, and he'd have at least 6 set combos and then 6 combos of (88). So 3 bluffs. If his range is all these pairs and Queens and no air, then I think you're good. He's just gonna be like “ahhh pot odds” and call.