Apply shielding paint to the cavities (a couple of coats, and check it afterwards with a multimeter), including the underside of the scratchplate and the control cavity covers, and check the earth from the bridge to the pots. All of your pots should also be connected in an earth loop.
I had a problem like this yesterday. I have to pull my SG apart again and put shielding in as the BKPs seem to have made the feedback worse, especially when I stand in front of the amp. Also I have a much more powerful amp now than what I had when I first installed the BKPs, so I didn't notice it then, partly because it was a combo low on the floor and thus the guitar rarely got close enough to do it