3/6 live bluffing heads up vs TAG

[DeathDonkey]

3/6 live typical great game. My friend is on my left and he is good, best at the table but I know how he plays really well and vice versa.

Anyway one limper and I raise AJo on the button, he 3 bets in the SB and its folded to me. I think about capping to throw him off but I call. The flop is T93 rainbow and he checks. He will checkraise me with any hand here so I check behind. Should I have bet anyway if I want to win this hand vs a probable bigger ace?

Turn pairs the Ten and he checks again. This time I bet in tempo and he thinks a bit and calls. River is another low blank and he checks. I can't beat anything he holds in a showdown so I bet again. Standard?

Btw I know he didn't play this hand well but I think he was experimenting against me.

Anyway thanks in advance for any advice,
DeathDonkey