$200 NLHE Heads-up: Shove or flat?

Villian Stats (VPIP/PFR/AF): 52/44/4.8

I rarely post HU's hands. A fairly aggressive dynamic going on. Villain is a decent reg for these stakes. I was playing him on a couple of tables, and I had some spots where I over shoved and I know he had a fairly big hand, lots of draws missed, and he wasn't paying me off. So I wasn't sure how often I could get him to pay off with Qx/Tx.

Hand Converted by Ace Poker Drills Poker Training Software

NL Holdem $2(BB)
HERO ($214)
BB ($296.35)

Dealt to Hero 2:spade: J:spade:

HERO Raises To $6 , BB Calls $4

Flop ($10) T:club: Q:club: 4:heart:
BB Checks , HERO Bets $8 , BB Calls $8

Turn ($26) T:club: Q:club: 4:heart: J:heart:
BB Checks , HERO Bets $18 , BB Calls $18

River ($62) T:club: Q:club: 4:heart: J:heart: J:diamond:
BB Bets $42 , HERO Calls $42

He can hardly put you on trip J, unless you think he has K9 why are we not raising for value here against a likely pairs of Queens or a busted flush draw.
If he has laid a few hands down already to shoves he might react and look you up with a Q or 2pair.

I think I explained above why. Considering my read, was it still +EV. Kind of hard to post HU's hands I think. Ran bad that day though so just double checking some things.

The problem is AQ/KQ are almost never in his range. He's probably CRing those a large % of the time. So that leaves Q9-Q2, and some Tx combos. How many of those is he calling with once I raise? He has to put me on Jx a large % of the time, and considering I understand his range and what he knows my range is, repping a busted draw is probably -EV here. I dunno.

I dunno much about your either of your ranges here but I would be flatting the river for sure. Shoving seems too thin all things considered.

I'm guessing the turn bet is for value, isn't it a bit thin?

I'm not sure if I'm always betting this turn, though I can understand why. As played, river has to be a flat. I doubt he's calling with a range that makes a shove +EV as his river bet is weighted heavily towards max value/bluff

I think in a vacuum river could be a flat but if you're bluff-shoving rivers often vs. this guy I can see this being a hand to put in our value-shove range.

I'm betting the turn, because I'm betting a lot of turns on this kind of board because of all of the river bluffing outs (and partially for value).

And river I am shoving/betting a ton of them, but I couldn't decide here. Being the results oriented person I am, he did have TJ so if I missed value, I'm glad I did here.

Thanks guys.