I like it up to the river. Villain will probably check back a majority of his Kx when the jack pairs so he can have lots of relatively strong hands to call the river bet. AT also has a reasonable amount of showdown value against his flop bluffs that give up.

Not sure if I should recommend this at 2nl, but... A line I like to take in this spot is x/r the river. If you check and he checks back you should win a fair % of the time. If you check and he bets he likely has TP at best and you can make a pot sized raise repping trips to get lots of players to fold that hand.

Given how the hand played out AT can still have some showdown value by the river and I don't think you have to necessarily bet it especially at NL2 where you might get some really loose calls.

This is a spot where I have been betting on the river with a hand like AT when a +EV check was possible. I still expect folds from hands such as 8x, and pocket pairs. The other thing is I'm not comfortable bluff catching with Ax here and expect a fair bit of random stabbing on the river.