I wouldn't bet pot vs unknown. Not a ton of value vs SB calling range.
As played I don't hate calldown if you bet something like 2/3, but because you bet big I think villain will be value heavy to bluffcatch effectively

NL2 pot = very strong. NL2 Click raise = very strong, and that after you bet pot (which in NL2 means you are very strong - and I advice you to no play like this unless you wanna have a very polarized range in this spot and continue with ur bluffs in most turns).

If you dont have any notes on the villain I would say I fold turn 95% of the time.

Population isn't going to be betting enough worse hands for value with this line and likely has a strong range once they x/r a psb on flop...You need to size down the flop as well..It's okay to use a larger sizing as you want to bet a more polarized range and force Ax to put more money in on flop, but you're rarely going to get three streets of value by starting off with a psb on the flop, and while QQ might want to use a larger sizing on this board there are plenty of other hands in your range that would benefit from using a smaller sizing to get more value from worse hands, like 66-99 etc.

I can get behind checking back flop or betting smaller. What do you estimate his calling range is from SB? You are ahead of all of his "non nutted combos". You crush all pairs, you crush all connectors, Ax and Kx are drawing to 3 ours and his Ax with a pair (A4, A3, A2) are drawing thin also. This makes me close to indifferent to checking flop or betting small. Checking benefits from cutting the tree to two streets which is great vs. his nutted combos. Checking also keeps his potential turn bluffs in so you can get value out of the weaker parts of his range on later streets.