The carpet and padding under it are most likely your biggest issue but I would still remove the cages for a bit and shim the legs as your floor is most likely not super level and no matter how good a carpenter is there is always variances in leg cuts. Level is very important. I bet you will notice a bit of an improvement if you do this. A pack of wooden shims at home Depot is really cheap maybe a couple bucks. I have 2 custom built wooden enclosures on a modified entertainment stand on very unlevel floor with carpet. A lot of shimming and leveling and it is rock solid now. I literally have a hump in the floor from an addition a long time ago that was done poorly.
That's an idea that I'll try. The carpet and padding are on top of a concrete basement floor so I wouldn't think there would be too much leveling needed but I guess it's worth taking a look at.