Don't know about bet-folding, only 54 got there and J6s(1combo). He as well can have some 22 and 36s.
You need to be right 30% of the time, and it is close (maybe some A4-A5spade, Jspade decide to rip it in and overvalue KJ), however those 54 (16 combos are very gross and likely without info we can make fold)

Why only a min-raise preflop ?
Why do you think the villain is very bad ?
I like your raise on the flop as his bet seems not to strong and you have top pair top kicker. It is a small raise and will keep almost all his paired hands in.
On the turn it is a tough decision as it is only 3$ more in an over 6$ pot so you need to good less then 1 in 3.
I agree with Jeff that folding might be the better option as there are so many hands that beat your top pair 3x 22, 3x 33, 16x 45 if you also count the unsuited hands. You already have shown strength on both flop and turn so villain is not that likely to overvalue KJ.

HIs range here on the flop is just going to be massively weak. He has 9 combos that beat you (if he has JJ or 32s here) and 450+ that you are crushing (like 85/15 or something like that). He has 4 combos of 54s drawing to 8 outs (16% to hit on turn). His range is super easy to play against here. You slow down on a K, Q or 6. His broadway range is not going to be heavy in QJ, or KQ or KJ here, its going to be made up of K9s, QTs, Axs and those holdings are drawing to 3 kings or 3 queens (very thin).

I am looking to get as much value out the weak parts of his range (the 85%) and keeping the pot manageable going to the turn where I am really afraid of so little.

I like raising smaller on the flop if you are going to raise. There are tons of weak top pairs that you don't want to lose to a fold. I am mostly calling flop here (in a vacuum) with AJ. QQ & KK are a much better raise here since he will have more AJ in his range, meaning he has worse hands in his range when stacks go in here on turn.

I am calling flop here and betting all non K, Q or 6 turns for 1/2 to 2/3 pot. Probably betting an A turn but maybe a bit smaller since it also fills 54s. The A isn't horribly scary though since he has some Axss in his flop betting range here so betting it certainly has value.

With specific exploitative reads here, your line is certainly going to be fine. I am describing a vacuum situation above and trying to estimate is BB call range here preflop, which I am sure is going to be super wide.