This hand happened recently at the borgata.
Hero has sat down for a couple of orbits, played few hands no showdown.
v1 has played few hands in this time but 3bet bluff a short stack from the SB with K3o and spiked a k to win.
v2 and v3 are in mp and loose but not very aggressive pre.

action:
v1(530)(utg) limps for 5
v2 (800)(mp1) limps
v3(1300) (mp2) limps
SB folds
hero(850) (bb) makes it 40 with AsKc
everyone calls

flop(162)
Ah Qs 4h
[we think V1s range is jj-88 AQ-A8s/AQ-AJo KQo KJs suited connectors down to 54s one gappers esp those with hearts. we think he would almost certainly raise utg /or 3bet with aa-qq/ak]
hero(bb) bets 100
v1(utg) calls
[ we think we can narrow his range to AT+ , TT+, KQ, heart draws plus floats hoping to bluff on later streets]

turn (362)
Ah Qs 4h | Jc

hero(bb) 150
v1 (utg) calls

should hero have bet here. (looking back not sure)
what if v1 raises? fold.

what do we do on following rivers.
hearts
non heart T
non heart K
rags

I'm really confused why we would think this. Limp-calling for 7.5% of effective stacks UTG with, say, ATs or KQo in what will prob be the worst relative position in a 4-way pot is just horrendous. Especially because we already know him to be aggressive (K3o hand), I don't think he's doing this ever.

In my view his range is extremely strong, basically just {JJ-TT,AK} and some slowplayed QQ+. I would be check/calling from the flop on, possibly folding to a river shove.

the only reason I'm thinking v1 has a wider range is because he wants to keep v2 and v3 in the hand who have been seen limp calling healthy raises.

Which makes sense if he is very strong, but with like KQo he's going to flop the best hand 4 ways maybe 10% of the time...

On the river the pot is 662 with like 240 left in stacks.

With almost any river card:

Check/fold getting like 4:1 seems bad.

Check/call seems far from optimal.

You flopped TPTK against passive live villains. Only one sticks around after the flop. I don't mean to be unscientific here, and sorry for not providing ranges, etc., but I think you might be overthinking this quite a bit.

The best lines here are:

Bet 120 on flop, shove turn. I think is the best line.

Or bet flop, turn, and shove river.

Just bet, bet, bet. Probably not folding to a raise given just the one hand we saw villain play with the Kx pre-flop 3-bet raise. If one of the passive villains raises the flop, then you can fold.

Pretty sure he's never floating in this spot (he was first to act after your bet with 2 others behind), but he can certainly have marginal combo draws and pair + FD. Bet flop a little bigger and jam the turn.