Villain seems regish with his betsizings. J does help his flop calling range, hands like J 10, J 9 and vs my flop sizing could definitely see K J, Q J aswell alongside A J. Still good turn to barrel and shove a lot of rives to fold out Ax and 8x?

PokerStars - $14.39+$0.61|10/20 NL (2 max) - Holdem - 2 players
Hand converted by PokerTracker 4

Hero (BB): 1,530
SB: 1,470 (VPIP: 50.00, PFR: 25.00, 3Bet Preflop: 0.00, Hands: 4)

SB posts SB 10, Hero posts BB 20

Pre Flop: (pot: 30) Hero has 4 5

SB raises to 40, Hero raises to 120, SB calls 80

Flop: (240, 2 players) 7 8 7
Hero bets 91, SB calls 91

Turn: (422, 2 players) J
Hero

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Kind of a close spot.I think that there are a lot of Jx type hands in his range.
Just because it's the start of the match I dont think that he would widen his 3-bet calling range.I dont see many low SC's in his range in this spot.
Many of the hands that he is flatting with now have a pair like 9T,J9s,JT,JQ(Maybe),KJ,AJ.
So I think i lean toward a one and done bet on the flop and X-give up on this turn.
Now if you decide to barrel turn you should shove many rivers especially the A and all the bricks that doesn't change the board texture.
Plus I think that even if he calls turna dn you shove river and the river improves him to a better two pair he is going to be reluctant calling a shove just because if you have an overpair his two pair is not good against you.
The downside is that you rep so little.