Phillip,
Thanks for the input. I will look into different subwoofers. I may have succumbed to the REL marketing.
Rel subs are for british people. IN britain, homes are small and close together and subs aren't supposed to make bass that would bother neighbors.
But if you just want the best sound quality you need
1) Low distortion (so that you're only hearing the fundamental, nothing extra)
2) High output (so that dynamics aren't suffocated)
the best way to get the above two is to go with large, low distortion, high excursion drivers, like the TC Sounds LMS 5400 or CSS SDX15 mk II. Funkywaves uses the TC driver in their 18.0 subwoofer.
The tradeoff for going sealed is low end sensitivity. You need to add some boost in the form of a linkwitz transform to get flat in-room response down to 20hz. Adding boost needs you need a lot of amplification. How much amplification you need depends on your max SPL needs, but it probably wouldn't hurt to have around 3000 watts
The best way to get this kind of amplification is with pro-amps, which are normally fan-cooled so they need to be in a different room to keep noise floor high. They're as good as any hi fi amp, but the fans are the issue.
Instead of sealed, Passive Radiators or transmission lines may be a better idea to get deep extension. PMC makes transmission line speakers, but they're so uber over priced i would stay well away. Funkywaves uses passive radiators in the FW 18.3.
But like i said, if I were you, I'd get just the Nao Note first and live with it for a while. It's a DIY speaker, but you can get much of the work done for you (flat pack enclosure kit, plus pre-assembled crossover). This should only run you around 3-3.5k for everything, so it's still half of your original final budget.