If you are slick about it the false wall you build can act as a bass trap. That would not involve using 4x4s...
The side of your false wall that faces the original wall can be a sandwich of sheetrock, particle board and sheetrock, and the other side can be a resonant membrane on the same frame. You'd want to not frame the wall on 16" centers (given no load to bear and the desire for the interior side to be free to vibrate/absorb.
Wes Lachot did a nice back wall design for my control room... might be worth having a designer help with the issue...
Oh, and you can get heavy sheet vinyl that is clear, so you can actually add mass to the existing door.
tom