min raise

Ask yourself what range he min-raises/4-bets?
For you to be slightly ahead he would have to be playing AJ+, 99+. Unless you have a read on him being very loose, I think this is an easy fold.

Note that villain is very short. The pot is 38 and we have to call 19. We need only 33% equity.

What if everything was the same in this hand but Hero was holding AKs? Never folding?

It is impossible to 3bet/fold JJ BB vs CO for 28BB. The only sensible option is 3bet/call and I would actually be pretty happy about my situation in this spot.

Folding is for me out of the question. Especially with a big 3bet like this.
I think 7-8bb in total is more than enough in a spot like this.
It’s indeed a little bit suspicious he isn’t raising more with 1 limper in the pot, but without reads we can’t do much else then stacking off.

Cheers,
SDK1987

Easy call for me regarding the pot size..

Although its already answered I'll try to connect the answers together in a hopefully logically completed way.

You're in an allin situation, so you can luckily analyze this situation by “just“ looking at your needed equity regarding the potsize against his handrange.
As la55i explained, you only need 33% equity. His handrange is difficult to predict against an unknown player on nl2 who plays minraise/4bet (especially after a limp as SDK1987 pointed out).
But you got more than 33% equity as long as you assume he would at least play AK as a nonpocketpair with this line (example for a pretty tight range: QQ+,AK -> around 36% equity). It gets better for every non pocketpair you put in his range.
The only reason to not call is if you put him only on pocket pairs (JJ+ or tighter), but I think thats really unreasonable because at least AK could be played like that as a “trap“ and thats enough for a call - not to mention AQs, TT or even bluffs.