Unfortunately I won't be putting anything up on the walls except paintings… And no stage setting either...
I already have a nice carpet under the kit, but it didn't do much. What I did yesterday was remove the bass drum's front head. I thought we had a blanket in there, but we had put a large, heavy, flexible pillow (the big ones you use on sofas for decoration). When we put it in last year I had it so the sides went up the left/right sides of the bass drum, not against the heads.
Once we removed it I had my son play the bass drum without it, and it sounded terrible (hollow with lots of vibration). Put the big pillow in and pushed it up against the heads and it was too muffled.
So we looked around and found his baby blanket from his crib (scary, that was only 10 years ago...!) made of some silk like material with probably foam in it. It’s big enough that we could cover both heads (to about the mid-way level). That sounded better, but still not fantastic, but not muffled or hollow either. I did tighten the front head a lot, but the batter side not as much. I had it looser but it didn't sound good, so I tightened it up a bit more.
We'll try that for now. But I'm open to suggestions on other things I could try inside the bass drum that might help with the sound.