I think your shove is fine. PLO is a game of aggression, being aware of your hand strength and how that can change. PLO is a game where you want to bet your strong hands and bet your strong draws, however not necessarily recklessly.
So to your question, should you shove with the nut straight on this turn. Yes, you have the nuts on this turn and they're showing aggression. You have to get your money in ahead in PLO. Without knowing the other players other two cards I can't get the exact equity. If the villain has K♥T♥XX against your hand you have about 72.5% equity. So this is a good shove.
There is one less J as you have one, so if he has KT like he does, he will only have 3 remaining, that reduces the combos dramatically. The fact we also have TT means there are also only 2 Ts left, again dramatically reducing the number of combos they can have for a nut draw. Likewise you also have a 9 so that also reduces the number of combos they can have to sets for a redraw to a fullhouse.
I think the shove is fine, they really can't have a lot of strong draws, and even when they do have them you're blocking a lot of the cards they need. Flush draw is always one of those things that's a factor but again you got it in rather good with a sizeable equity amount.
To me the mistake you made in this hand is playing so passively pre-flop. Mid position with JTT9 can be an easy raise. It plays much better with less players. I think your pet on that flop into that number of players is probably overly aggressive. You hate when you're re-pot in this case, because what's going to re-pot you is a set and you're going to be in a situation with a draw where you need to decide fold or commit entirely to the hand.