I had the same problem with my BK-9. I lived on the second floor in an apartment and I used to practice late at night with my headphones on, the neighbor below me came up saying that he is hearing repetitive noise on his ceiling, and he can't sleep, 'cause it's annoying him. He was right, it was loud, especially when there was total silence at 2 AM.
So I went to Home Depot, and bought some of those wide foams with a sticky side that they use for window insulation (weather stripping foam seal for doors or windows), I put two layers under the white keys (no disassembly required), pushed the foam in gently so you can't see it, and this way you can't even tell that it's there, and it did quiet things down, couple of notches.
Easy simple fix, although the black keys have no damping underneath, but it seems OK for the white keys. And also playing is easier, feels different, although it depends how high you stack those foam insulator strips.
Although make sure the glue on the sticky side is not that strong, so you'll be able to remove it, in case you don't like it.