Used to be they had think felt-ish pads for under typewriters, to isolate the machines from the desk tops. Cut down on noise in the office. Don't know what similar is out there. But back in the early 1960s on the wrestling team, the matches were held on slow footprint pads about 2" thick. By setting your UUT ON something like those instead of suspending, might be simpler?
For rubber suspensions, maybe something heavier, more like motor mounts from automotive land. In any case, seems to me it would be better to use compression rubber mounts rather than suspending things with rubber.
I used to have a couple wheel tables in the shop. A roughly two or two and a half foot square table a bit lower than bench/counter top. And on wheels, so I could roll it around. Set a Fender Twin on it, and I can sit there and spin it front to back as needed. Then pull the chassis, and set chassis on bench. SO instead of some big dedicated piece of furniture, consider a smalll piece of plywood say 18x24" and bolt your buttkickers to the underside. This would be large enough to sit an amp on. Give the board a few inches of legs to clear the buttkickers, and some sort of padding to isolate from the table top. Now, you get this thing out only when you need to run this test. Set it on a wheel cart, or on an empty bench, or wherever you have some horizontal surface to use. Otherwise it is out of the way.
Not sure I like your test signal. Scope says 20ms, and I see roughly two cycles per grid, so 100Hz? Modulated by 8-10Hz? The 100Hz seems un needed. You know what occurs to me as a signal? A drum machine set to a couple patters. Crank that through whatever drive amp you had going. That seems like a fairly reasonable test signal for the amp as it mimics music and the sorts of vibrations on stage.
Sounds like a place for an old Crown 300 - those worked all the way down to DC, hence "DC300".