I'm going to start my answer with a question to you, by shoving $800 to win $200, what calls you that you're beating? I personally think the answer to this is nothing calls you that you're beating, maybe 10s. Why are you risking so much to win so little?
In general, good LAG players are very self aware of themselves, how the table are perceiving them and whom may be fed up and taking a stand against them. This is important as if I was that LAG player, I'd check to you with all of my two pairs, all of my sets, all of my pocket pairs above 10s. Just to stress it again, I'm taking about good LAG players here.
I'm not saying betting your JJ here is bad, far from it, I think you have a lot of value on that board and should make the LAG player pay to see another card. Hands like 78 here would totally love a free card, but we should charge for that card and protect our value. However with that said, I am saying what is your $800 bet achieving that a $125 to $250 (some like to overbet) bet doesn't achieve. I believe the answer to that is it achieves the same thing, but also protects you from going broke when the LAG player has aces, a flopped set, or a funky two pair.
You mention you've seen them draw, but you've not mentioned the context in which they've called on those draws. Someone betting $25 into a $100, yeah I'm also going to draw to my open ended straight there. Or someone betting $50 into $150 dollar pot, and if I have a pair + overcard + showdown equity yeah I'm probably going to also peel a card there too. Similarly if I have an overcard and a straight draw, yeah I'll probably peel too if the price is good. LAG players tend to not call draws without huge equity here. Good LAG players like to dictate the action rather than passively call. By you shoving you take away the ability for them to drive the action.
In short I don't believe this is a positive EV play to make, as you're only going to be called by better, and we can put ourselves in a position to be able to fold on a later street if a disaster card comes. Basically the LAG player isn't going to call all-in with nothing, and likely wouldn't call with nothing with a $125 to $250 bet either, so why risk it all when you'll only be called when you're crushed. If they do call this bet with nothing, well congrats you have something you can exploit now too.