General River Concept Question

[Lunar Tweak]

i find myself in this situation mostly when i'm in the blinds and out of position on the river. here's a hypothetical hand to illustrate my dilemma.

Hero has KQJT and the board on the turn is 4 9 8 T. Hero bets and gets 3 callers. River pairs the 4. Should Hero bet, or check/call(or even fold)?

does the answer change if:
the river pairs the T?
Hero get's more or less callers on the turn?
there's a low possible?

My thoughts so far are that you should bet if you can fold to a raise, meaning no low possible so only a Full House will raise you. otherwise check/calling is probably the way to go, but i'm not certain.

also, anything else to consider when this happens on the turn? maybe simply check/folding more often since the pot will be small and not usually worth pursuing?

thanks in advance