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Well, assuming he's on, say, Jc Jh, probably the most dangerous hand, you're 34.75% to the river, which makes the push about even EV-wise after his raise, assuming you have no FE.

If he's on, say, JhTh, you're up to 44.5% to the river to win, and if he's on AhJh, you are a favorite to win the hand and the push is pretty much mandatory.

When he minraises, you're getting over 4:1 to call, so that's a clear call even without taking into account any sense of implied odds. You should almost certainly call here if he offers 3:1 or better. If you opt to smooth call and you don't catch a K or diamond and you check the turn and he moves in, you'll only be getting 2:1 and be a 3:1 dog to a made hand and should fold unless you think your opponent is on a draw.

That's really the problem, of course: if you don't push, you give a weaker hand (say, 89d) to try to take it away from you.

Since you are getting >4:1 to call, you must call. Your EV from calling is around .25*(the pot after your call), or about 275-300, plus (.25*(whatever you can get your opp to pay off)). Say you feel on average you can extract about 400 more chips if you hit. That means your call is worth 400ish.
Minus your cost to call, that's about 180ish.

If you push, it depends on what hand you are up against, of course. Your EV is something like

((your odds of winning * total pot if he calls) - (1200)) + (odds he folds * 710).

Since this is an 11, let's say there's a 30% chance he's on TT/JJ, a 30% chance he's on TJ, a 30% chance he's on AJ, and a 10% chance he's on something weaker like 89d. I'm just doing this in my head, but say that's around 41% total chance for you to actually with the hand against a caller. Let's say he folds 25% of the time, often with a very weak holding, and other times folding something like AJ because he realizes he's either a small favorite or a big dog. So our formula becomes:

((.41 * 3000) - 1200) + (.25*710)

If the hand ranges are correct, you're about ev neutral and your value comes only from the FE. So your +chip value is about 175.

So a call and a push are both +EV, and if you buy into my estimates for hand ranges, fold equity, and the amount he pays off when you hit, they're about equally +EV, in terms of chips. I'd say if you think you're a better player than most at the table, then you probably want to just call and reduce your variance, as the big stack you get from winning in a push confrontation is not as important to you, and the calling method does not risk busting out early.