I find myself in similar spots allllll the time with jacks. I would have shoved as well. still hold plenty of equity vs AK AQ type hands. especially from a LAG who could be 3 betting and calling off lighter than some others. at the end of the day, no correct way to play JJ lol gl gl in the future.

I think both jamming and flatting the 3bet are both reasonable. LAG is a bit of an ambiguous term, but in general vs. players with wider 3bet ranges I think pushing JJ into your shove range as the bottom of your value pole seems fine/good.

In general, I personally tend to flat JJ here and my value jams are QQ+ AK, then I also jam a bunch of weaker hands that aren't strong enough to flat. 99-JJ and probably 88 just slot nicely into calling the 3b most of the time. When I do shove JJ in this spot, it is generally done as an exploit vs. players who have wider 3betting ranges.

I don't have a super strong grasp of what GTO preflop ranges look like for this spot, and this may be a hand that gets pretty heavily mixed. Seems fine to go either way with this combo

This is monker response LJ vs HJ 3B 40bbs deep fwiw: https://gyazo.com/a07fcc7a8034fa13206600a679ba7f7f

I would not focus on this type of spots tho since at the end of the day you have JJ for 40bbs and if he has you beat is just a cooler. Vs some nitty guys could make a case that the EV of flatting goes up but imo this is not where our EV will come from

JJ doesn't play especially well postflop, jamming is higher EV. I think that particular hand is a clear jam. The adjustment you might want to make given he is a LAG is to pure jam TT rather than mix and pure call AA rather than mix.