A subwoofer that meets all your criteria has a new home in Germany. Audio Reference purchased California based Velodyne Acoustics. Now marketing some new models and assembling a world wide sales and service network.
While it may be more than you’ve considered for your budget The Velodyne Digital Drive Plus can serve as an incomparable master control for less expensive slave subs in an array. The Plus has the ability to control a daisy chained system of units regardless of their inputs. The Plus offers one of, if not the most uniquely easy to use and remotely controlled, subwoofer dedicated Room Optimization / Equalization programs.
After Auto EQ has completed, on pages 10 and 11 of the DD Plus Interface Manual describes how to use the Frequency Response and Parameters Screen. This is a simple drag and drop control of multiple parameters that are visually and audibly made at the listening position to closely match or augment your main speakers basic presentation from 200Hz on down. https://www.velodyneacoustics.com/pdf/digitaldriveplus/DD+UserInterfaceManual.pd
Compare the flexibility the rear panel offers to other subwoofer panels in the User’s Manual below
http://velodyneacoustics.com/pdf/digitaldriveplus/DD+Manual.pdf
I assembled a four sub asymmetrical array in my largest 26’x 35’ room and they did eliminate ALL the rooms modes. I later settled on two asymmetrically located slaved Plus’ which allowed a noticeable mode in an unused corner of the room. The room loading and intensity has been easily maintained with just two subs ever since. Point being you may not require as many subs as you may think you do. What ever you decide on take it slow but keep it low.
FWIW, ["Virtually no music is down in the 20-40 Hz range"]?
Whether my Bass playing skills would be deemed musical is certainly subjective. Depending on my tuning and without the OctaBass on my Lakland’s stainless low B rings significantly at 31Hz which most 10" subs can easily output. Those B and E strings are bread and butter for a Bassist.
