Based on your read on villian your A9 is ahead of his range, however it is not a very strong hand, if you 3bet and pick up the pot right there it is a good result, if he does not fold you have a better chance of picking it after the flop since he is less inclined to bluff after you reraised.

On the flop you got a relatively good flop for you, 2 is probably not in his range, so he either missed the flop, have an A with a worse kicker or has an A with a better kicker or has a FD, he can bet any of those

If he has a better ace we are in a bad spot, but while it is possible it is not that likely.
If he has an A with a worse kicker you are giving him a chance to catch up, as any A,2,K,Q,J,T spilts the pot and he can still hit his 3 outer to take the lead, so raising and getting him to fold is good, raising and getting him to call is not that bad
If he has a FD you want to charge him to draw
If he is bluffing then calling is the correct move hoping he would bet again, which is possible but not guaranteed.

I also think your fold on the river was bad, your hand looks like a FD more then anything else, he could bet all the way here with A with a worse kicker and with any mid-high PP