Something has to be off about the room, like you are sitting in a null considering your comment about the need for more bass. Have you taken any measurements at your listening position? Have you played around with speaker/sub placement or shifting the listening postion?
I run a pair of Focal Kanta 2 and one 15" sealed sub in a room twice the size of yours and have no issues with output unless I’m going for rock concert levels. I’d take some measurements and play around with positioning...speakers and listening. You might also want to look into room treatment. In another room, adding bass traps greatly improved the bass response of that system.
Regardless, I have a hard time believing there are no speakers in your budget range that can give you what you want when paired with a REL array. So long as you have addressed cancellation effects.