I can well imagine you are all over the place with this. I would be too. The 10x20' room is a tough space to do really well, and is it worth the investment when maybe the bigger
space could be a short term room without a bunch of features. The thing to me, is are you really stuck with the small space, in the short term. If it was all I had, then I would use it,
but the larger space built to a tight enclosed space might be two layers of 5/8" drywall.
Maybe some bean bag chairs for the kids might be a fun option for the short term? (Riser-less for either area for now in the big room?)
Ideally you want at least 3' off a back wall, and I would weight the opportunity of doing that, against the impact on the rest of the basement, since this is really just a temporary
space. The bit of rework plus waste of time and a bit of money, kind of steers me to the bigger space right off the bat.
I didn't see those Harbor freight mats on their website. But I doubt they would be appropriate to use with GG. I would expect two layers of 5/8" drywall with GG would be the
better performing wall, when you look at wanting similar mass of either side of the GG. Once in a while, some experimentation pays off, but most of the time, you end up with
an underperforming final result.