A few people have mentioned various things. Here's a checklist that worked for me (I had this problem extensively and have a similar setup to you).
1) Drop the pickup height away from the strings. Yes, it's lower output. Do it anyway, it will help massively with the feedback problem.
2) Lower gain on the amp.
3) Get a noise suppressor/gate - you've done this too.
4) Put rubber/foam material behind ALL springs - including those in the pickup mounting ring. I found most recently that the springs either side of the pickup were vibrating and making a hell of a noise (because they're close to the pickup so were getting amplified too!!!). Same goes for springs of the bridge, and also worth damping the strings behind the nut - I found that they were ringing a LOT, especially with certain strings and not others, and this was adding to the problem.
Hope these tips help. But yeah, drop the pickup height a bit :)
Roo