You're on the right track, but not quite. It's not the euro style exercise alone, it's actually the contango of the futures contract (presumably the very reason you're in XIV). But they don't have true 'intrinsic' value when the VIX goes 'ITM'--in short because there is no value that underlays it. But there's an expectation that this will revert back down prior to the June expiry--the only way that VIX options will exhibit price movement vs. the underlying similarly to equity options is when the expectation is that this will not come back down prior to expiry (i.e. people believe it will flat line from here).
All is not lost though, as long as you can swing the margin. If you don't mind holding this until June, you can wait till it reverts to lower values, and if it doesn't your calls will become worth something as they near expiry, and on expiry should be very close to you expected value.
I suspect that the trade will be a total loss from this point anyway since you're locked into it, and anything you pick up on XIV will be offset by the premium losses on the long calls...but you'll do better than trying to close it while the VIX has run away from you.
Chalk this one up to a lesson well learned for relatively cheaply.