It just so happens I've been 'helping' a friend try to 'spec' a new modest HT system, and it's come down to the same topic.
(But with no good candidates, either.)
Also, I've recently made some upgrades, and have changed my thoughts about 'full range' & subs...so a timely post.
I'm finding a lot of tracks that have a surprising amount of content below, say, 40hz; it makes a Huge difference to have the main speakers handle that, esp cleanly and sharply. (Can also be heard, even at low levels, at the other end of the house!).
Granted, most of those tracks are new ('born DSD'), organ or electronica, but just normal bass and drums 'have it' too, if they are well mic'd. (Pat Barbers' Café Blue 'unmastered' CD, for example.)
SO. My new thinking is to aim for full range main speakers that can do an 'honest' 30hz, and look into subs (servo) that can do the power reinforcement and extension from, say, 10-15 hz to say 50hz.
My current (non-servo) sub doesn't go that low, but does have the reinforcement thing down pat. I've got it set now to just 20hz - 45hz.
(PS Audio has a few useful YT clips about servo-subs.)
Problem is finding full rangers that will go that low, cleanly and loudly.
JBL M2 and 4367s will do it, but out of your budget range.
PS. I have GE centerXL and surrounds. I like them very much, and are extremely well made.
I listened to one of their early towers (I think the 3rd gen top of the line, not the current TOTL) and was reasonably impressed, but wasn't thinking about sub integration at the time. Also not horn speakers, which I have a fondness for.
Good luck - interesting challenge.