$1/$3, 6 handed

V ($275, BTN): Weekend regular rec player in his mid 20s. No history with Hero but V knows he's a regular. Been at table since 30 mins, very chatty and said earlier he has to leave soon, etc. Saw him play a hand in his first orbit where he called a $10 EP raise in MP where it went 5-way to a flop of JT7. The OR bet $35 and he quickly announced $100 (with $100 behind) and flipped over 77 by mistake when action was back at the OR because he thought OR was all-in. He seems to limp quite a few hands pre because he'll probably be gone pretty soon.

Hero($380, BB): Been playing fairly snug and haven't been involved in a hand since V got here.

Onto the hand...

HJ limps, V limps on BTN, SB completes, Hero raises to $20 in BB with JJ, only BTN calls

Flop ($46): T65
Hero bets $30, V looks at me asks "how much was it preflop?" and quickly calls

Turn ($106): 2
Hero bets $65, V quickly announces $165 (with $50-60 left behind), Hero ?

Sign call might be Ace 10.

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I used to find myself in these situations often. Situations where I had an OP to the board, and I'd find myself leaking money far too often.

This looks like a fold. Have you seen your V play any of his big draws? Unless he is the type to play 10x of hearts or 78h this way, I can't find anything you beat. You've showed strength on three consecutive streets, and he still doesn't care. Could be a slow played set or a higher OP. (You did say he was passive pre.) But it sorta smells like 34, more than anything.

Now like I said, if your V is spewy/tilty, I can advocate continuing in the hand. But the vast majority of players at low limits aren't playing around in situations like this. Unless there is info about V you didn't tell us about, or if he's known to fast play his draws, it's a fold.

I'd need more info on V before I'd think he's limp calling 43, esp when pot is shaping up to be HU. Maybe 43s but that's a stretch. Not having more info on his limo call range is an issue here. We know he's limping a lot but what is he calling with if action is raised? 65s, 43s, all pairs?
If his range is only AT, 65 and 43, then JJ is 35%
If his range is AT, 65, 43, 77, 88, 99, then JJ is 54%
If his range is AT, 43s, 65s, JJ is 60%.
There are other ranges you can construct here, but just thinking about his most likely hands and how he'd play them pre flop and post flop, I think AT, 65s, 43s make the most sense, esp given his previous action where he eagerly raised his set on the flop. But even if you give him 66 and 55 in this range, JJ is still 47%, which, with the dead money in the pot, makes this a call.
I don't think stacks are deep enough to fold. I hate doubling up a Ayer who is about to leave, but if he's overplaying hands bc he's playing within a limited time horizon, this is the chance to get his chips before he wriggles away.

you just barely beat top pair. so you decide if he will raise without being able to beat top pair. that is the decision.

Fold. (WP otherwise)

if you know something about the guy then go with your instinct. but most times when someone plays like this they can beat top pair.
you can give him whatever range you want to, but when you get to see his hand it will beat top pair.

He's telling you that one pair is no good. Most players would have raised top pair on the flop rather than wait for the turn to do so.