I can't imagine anything short of a pair of SVS 20-39 PC+'s would fill a room that size with SPL close to reference level, and that's already double the budget.
I'd also reiterate the use of a good tactile transducer(s) coupled w/a 20-39 PC+ or something comparable. This would still likely be a few hundred dollars more than the allocated budget, but would bring you much closer to what you're looking for in this area. -TD