Preflop: Hero is SB with A
Q
3 folds, Hero raises, BB calls
Flop: (4 SB) 4
2 8 (2 players)
Hero bets, BB calls
Turn: (3 BB) 5 (2 players)
Hero bets, BB calls
River: (5 BB) J (2 players)
Hero checks, BB checks
Should I bet river against worse A highs? If I check I think he will bet all better hands and check A highs but if I bet he might call with them. From what I've seen he is straight forward TAG
His stats vpip/pfr/af/wts: 31/25/2,1/41
If you have to pay off 1BB anyway, i would bet myself as you suggested. The reason that you cannot fold is that he could have some x3 x6 hands or 7x hands like K7 Q7 he peeled again with on the turn so i don't actually see myself ever check/folding this river. 8x he would have raised i think and he may call with A6 A7 A9 and some Kx type of hands and i agree on your suggestions that he's almost betting any better hand so bet/fold would be my line. 
