Flop a boat, how to play the river agaisnt definite trips or better

Empire Full Ring $100NL

Hero has $100 and Villain has him sligthly covered. Read on villain is that he’s pretty loose, but not entirely crazy. His play is pretty transparent. I’ve been playing with him for a couple of hours, and twice I’ve seem him smooth call the flop and miniraise the turn when he flopped trips or a set (one time with trips and one time with a set).

Hero has 7c 7d in BB.

Preflop: folded around to CO who calls, Villain on Button calls, SB calls, and Hero in BB checks.

($4) Flop: 7s 10h 10s

SB checks, Hero bet $3, CO folds, Button calls, SB folds.

($10) Turn: 7s 10h 10s Kd

Hero bets $7, Villain raises to $14, Hero reraises to $45, Villain calls.

($100) River 7s 10h 10s Kd Jc

Hero???

OK, here’s my question. I have $51 left. I’m sure Villain has at least trip tens, but I didn’t really want to see the turn and river both be face cards. My read on Villain (from two hours of play with him) is that he’s got a 10 with a J, Q, K, or A kicker. There’s also a chance of 10-9s or 10-8s, but I’m betting on some kind of face card more times than not. I’m thinking there’s a 50/50 chance he’s filled up on me. So do I just push the rest of my money in here?