I agree that Caribbean has unbelievable value that time of the year...
Let's take Amanera for example...You can book in June for 5 nights or more under Season of Discovery and starts around $1,040++/night. Sure, it's more than the 800/night budget (are you referring to EU or USD?) but it's inclusive of full board so you're probably not too far from it but I think tax is another 27% or so, though. Point being, great value in the Caribbean.
No good cities but maybe just stop in an EU city along the way.