I'd bet. Given that it's 4 handed and you haven't shown much interest I doubt he's going for a x/r here and I think the small number of worse hands he can have will call.
Seems like a freeroll bet to me. His check is weird if he has you beat. I think the most likely worst case is you chop with him. I just bet in these spots and sometimes get called and win. I stopped being shocked by what they sometimes call me with.
I, too, would bet. I wouldn't worry too much about losing but what he shows will be telling. Fwiw I can't think of a hand that he can show you that will support the "competent" part of your read. Best he can do on that front is prolly something like 76dd.
There are folks that I would describe as "semi-competent" (better than most but still not good enough to win) that would donk this flop with 8x, 66,77,99, or even higher pairs that wanted to slowplay but got lost when an ace flopped. Then they bet to "see where they're at." Upon being just called, they have no idea where they're at and just sort of stumble through the rest of the hand.
Therefore, I would bet.
I bet against most villains (including someone I’d describe as “semi competent” at these stakes). I’d check against really weaktight opponents and opponents I’d describe as very good. If I took villains line against someone with a brain it would usually be with the intent of check raising the river.
The rationale for betting this river should be trying to fold out ax hands we chop with as villain is not ever folding better and will sometimes xr (rarely a bluff)
I don't think similar weak ax hands fold that much given hero's line though
So I think this is a fairly straightforward check
Edit: I am defining semi-competent as closer to a good than bad player, if villain routinely butchers mid-pair type hands going for value is a lot less thin
Not sure hero really has many hands to bluff here but if he goes for thin value with a wide range bet looks a little better
The rationale for betting this river should be trying to fold out ax hands we chop with as villain is not ever folding better and will sometimes xr (rarely a bluff)
The rationale for betting the river is to get value from a poorly played KK/JJ/TT/99/8x. And maybe some random Qx bluff that got there and then got confused.