A long time ago, I read where Dave Weckl gave some pretty good advice re: overbearing sympathetic snare buzz. @ least it does work for me.
According to what Weckl said, the 12” tom is the biggest problem on the kit in regards to that problem. He said that the snare drum is ringing & buzzing b/c there is a harmonic from another drum setting off the snare drum.
Hit the snare drum around all of the lugs & you’ll probably find 1 or 2 lugs that are just a little out of tune. By tuning them up, your snare buzz should dissapate.
Also, the advice on using more quality snare wires is great advice, as well! Puresound Percussion & DW’s True Tone Snare Wires are my 2 faves!
I really hope that this helps & keep us posted on how it works for you!