Now that you've upped your budget I think a used pair of Joseph Audio RM7XLs would be very worth consideration given what you're looking for.
Also, I don't think tripling the wall thickness will do much for your bass problem although it may help a bit with the mids and highs (especially if you use something like green glue between the layers). The walls will still be attached to the studs, which will still vibrate with the bass and transmit to the wall on the other side. Plus, depending on construction the beams in the floors and ceilings will also continue to transmit bass to the other room. Unless you're willing/able to install something like resilient channels in your walls and, significantly more difficult, also isolate the floor and ceiling I don't think you'll be able to attenuate the bass transmission enough to allow you to crank the music very loud if it contains significant bass. Sorry to be a bummer but that's my understanding of it and don't want to see you waste $$$ on something that won't work, but maybe others have suggestions.
Also, I don't think tripling the wall thickness will do much for your bass problem although it may help a bit with the mids and highs (especially if you use something like green glue between the layers). The walls will still be attached to the studs, which will still vibrate with the bass and transmit to the wall on the other side. Plus, depending on construction the beams in the floors and ceilings will also continue to transmit bass to the other room. Unless you're willing/able to install something like resilient channels in your walls and, significantly more difficult, also isolate the floor and ceiling I don't think you'll be able to attenuate the bass transmission enough to allow you to crank the music very loud if it contains significant bass. Sorry to be a bummer but that's my understanding of it and don't want to see you waste $$$ on something that won't work, but maybe others have suggestions.