I've used a REL R-205 with Rega R5's very successfully. You should be able to get one for right around your budget.
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Subwoofers in general are not boomy -- rooms and poor setup lead to boomy. Pick any sub from any reputable manufacturer and be willing to spend the time on proper setup. Your room is small so even a 10" driver unit would probably suffice. I'd lean towards having PEQ somewhere in the bass chain. Your budget really doesn't allow for a separate PEQ device like the Velodyne SMS-1, so I consider any of the Velodyne subs that have PEQ builtin like the SPL 1000R
Following up Bob's point.
At Audio Advisor, the SMS is $450 and a sub from NHT (can't recall the model designation) which features dual 10" drivers in a 6" deep sealed cabinet is available at $300. Total = $750.
Caveat : I HAVE NOT HEARD THIS SUB.
Clean, deep bass (< app. 35hz) output will -I assume- be restricted by the subs' small cabinet design, but the shallow box should allow very smooth response when placed against a wall. The sealed design will likely produce highly damped "tight" bass. The terrific low pass x-over in the SMS will allow seamless integration with your mains and the PEQ will allow you to smooth and extend the sub's output.
One proviso, the digital high pass in the SMS isn't my favorite part of this device. It subjects the entire signal to A/D/A conversion. You may wish to try your main speakers both high passed by the SMS and full range (bypassing the SMS in the main signal path) to see which approach you prefer.
For most music, at less than head banging spls, this should be a good solution within your budget. If you later decide that you need "more" subwoofer (bass <30hz or more volume capability), your investment in the NHT is sufficiently small that (presumably) you won't get killed on resale.
As long as you system stays in a 10' x 10.5' room, you're going to have deep bass issues - if your system produces deep bass. A truly potent sub may well cause more problems than it solves. However, I would think that bass EQ (like that provided by the SMS-1), should prove extremely beneficial to such a set-up.