Advice is worth what you pay for...
In my mind, I'd rather have a structural engineer give me a piece of paper that I can hand to my insurance company, than a "verbal promise" from a sub-contractor.... call me silly.
But, if you're good with it, you want to cut the 5/8" FR gypsum just about an inch smaller than the cavity. Center it in the middle of the cavity. Use a coupla' drywall screws to snug the gypsum to the floor. Next, lay a bead of acoustic caulk in the bottom of the gap, and stuff that ~.5" gap with 3/4" backer rod. Once it's in there, seal the top with another bead of acoustic caulk. (I HIGHLY recommend OSI's SC-175 caulk for OH work!)
If you feel brave enough, lather, rinse, repeat a 2nd layer of gypsum.
Next, take a 1x2 furring strip, and cleat that gypsum in place. Caulk the seams along the furring strip, as well!!
Why do all this caulk, you ask?!?!?
For a VERY good reason, actually...
Wherever air goes... so goes sound!
You can't actually make a room completely air tight on your budget, but the more you can stop, the more isolation you'll have.
After you deal with the ceilings, and the walls... yes, you'll need acoustical treatment... but for the life of me, I seriously don't think you'll need nearly as much high frequency attenuation as you will LF attenuation (trapping)... so thankfully, you probably won't have to use any of that nasty looking foam crap that costs a LOT more than a nice piece of silk wrapped around a wood frame.