No soup for you...
I put my 70lb subs on extra soft rubber caster dollies topped with a two inch layer of open cell foam which greatly reduced the the low frequency transmission through my homes structure. Decoupling the subs and the substantial recalibration needed actually improved the subs low frequency presentation.
You'd satisfy your neighbors but a DIY cabinet with signal processing and a couch with futon like space might take you way over budget. There are some in wall or contractor models that may give you some ideas.
In my experience I would never consider 'sound REL-like' a goal. Good luck with it.