Personally I don't think I would three bet the flop unless I previously noted that button was a loose-aggressive player. Since the third player called two bets cold he is not going to fold if it is raised or capped.
I think you played the other streets fine. You are not going to loose the third player if he has A2 after you pick up the low draw.
Some players might consider betting out on the turn, hoping the button will raise and the third player will fold if he has A3 or A4. I don't think this is a good play. There are two many things you need to be just right for this play to work:
<ul type="square">[*]the button has to raise [*]the third player has to have A3 or A4[*]the third player would have called one bet but not two [*]you need to make your low on the river[*]the button needs to not have a better low[/list]When you multiply the probability of all these events happening together, it is pretty small. You will do better in the long run just putting in one bet on the turn.
Calling on the river seems like a non-decision. You will win either the high with your trip Jacks or the low with your second nut low the vast majority of the time. The button would need A2KK specifically for him to be ahead both ways (baring him raising the flop with a very week hand and backdooring a straight - a hand like A24K).