Wow, tough situation but an extremely interesting one. When the BB calls the pot is now 4050 and you need to add 600 to call, so the pot is offering your jacks
odds of almost 7 to 1 and 100 pounds of implied odds since the BB has huge stack, which means you would never fold even if you think you're dominated, hoping to get help from the flop. So, to state the obvious, you have to at least call and the question now becomes if you should reraise or shove.
Reraising also seems out of question here because it would likely make you pot committed, so shove or call are the only 2 options.
You mentioned that the BB is very loose and aggressive after the flop. In this situation I would be very cautious when trying to put him on a range for the reason that he called an all-in raise, where he knows that in order to take this pot down he has to show the goods, he can't bully his way into winning it, so we have to give him credit for some kind of hand. Here is where knowing your opponents tendencies becomes critical.
If you make him capable of trying to trap you with a hand like AA or KK, then you should just call, since it is possible that he has one of those
hands making you a 4.5 to 1 dog. If you are strongly convinced he would have reraised with one of those hands, then shove. He's likely then to have AK or AQ or some other hand that you are ahead of and you can probably get him to fold it leaving you HU with the all-in guy, which is the best thing considering the size of the pot already.
I'm not exactly a fan of JJ because there are too many ways to lose with it, but that is exactly the point here. If you think that shoving has a great chance to make him fold then you should definitely do it. It all depends on how tricky you think that loose player is. You don't want to get knocked out but you do want to play this in a way that gives you the greatest chance to take down this huge pot.