How tight is your tom's resonant head? You didn't mention changing that tension. Maybe try tightening that. I've had that situation, and it seemed no matter what I did, it buzzed. 

Buzz is caused by a tom having the same fundamental frequency, a harmonic of that pitch, or some other sympathetic pitch, as the snare side head. I personally don't tune any of my drums to specific notes. I just tune to a tone I like. I have found that doing this doesn't cause me too much of a problem with buzz.  I would try chaning the tuning of either the snare side head or the offending tom.  Obviously the snare side head will affect snare response so a combination of changes to both drums might be what is needed.

Another thing that makes it more of a problem is a snare side head not seating properly over an narrow and deep snare bed of an older drum. I use a hair drier to carefully shrink the head so it conforms with the snare bed so there are no wrinkles. Probably not a problem on a Supra with their very long and narrow beds, but I wouldn't rule it out. 

I have also found that the orientation of the strings on the snare end plates can make a difference. The strings going into the holes from the top surface so they run between the snare end plate and the snare head versus the strings inserted from the bottom and ending up on the outside of the snare end plate, thus pulling the plate more into the head can help.

I have also heard of loosening the tension rods on either side of the snare beds can help and I suppose it has helped, but if it causes wrinkles that usually makes it worse. 

I really have had no silver bullet, but these are the things I have played with that usually help. But, a little bit of snare buzz is often impossible to completely eliminate and is just part of the experience of live music.

Thanks much, Rik and Thin Shell....I'll try your suggestions.  For sure, switching the orientation of the strings thru the end plates is something I hadn't thought of.  I'll try it.  

FWIW, I'm using the "Evans Pure Sound Equalizer" snare wires with the "split strand design" (2 groups of 8 wires each, mounted side by side on the same end plates for "decreased wire vibration..and reduced sympathetic buzz."  They do seem to help reduce buzz somewhat, but they are not a magic elixer, for sure!

Regards, mb

To wrap up this thread, I did manage to reduce the snare buzz on my 12" tom, using various suggested remedies.  It's not perfect, but it's MUCH better than it was, and so - for that - I am thankful.  I didn't find a singular "magic bullet" that I can identify, but these are the remedies I tried, and I suppose they all contributed somewhat to improving the sound (less snare buzz) on my 12" tom...but it was all "by guess, by gosh."  FYI to VDF members: 

1.  I tried tightening the reso head on the tom.  No luck.  I only got a higher-pitched nasty buzz.
2. I put 13" Pure Sound "Equalizer" snare wires on my 14" Jazz Fest. I've found that the metal end plates of 14" snare wires touch the bearing edge on both ends - throw off and butt - but the 13" metal end plates don't touch the bearing edges, so I considered that a good thing.  The 14" snare wires fit a metal Supraphonic much better.
3. I tried loosening the tom tension rods closest to the snare, but no improvement there.
4. I put a Ludwig C1114 snare side head on the Jazz Fest.  That's a clear, very thin head that is recommended for use on a Ludwig Super-Sensitive snare, which I no longer have. That seemed to help.
5. I lowered the tuning on both my Jazz Fest and Supraphonic snares.  Drum dial pressure of 88 on the batter head, 78-80 on the snare side head (78 being the two lugs on either side of the snare gates.) That still gives me an F3 note on the tun-bot, for both snare drums.
6. I cut a business card in half, and put one half under the (snare wire) metal end piece on the throw-off side.  It is under the metal end piece and extends to the bearing edge.  That seems to help, but too much business card really chokes the drum.  Be sure to loosen the throw-off slightly, as the business card effectively "tightens" the snare wires.  I may or may not use that "live."
7. I adusted the tom to about 85 on the drum dial, both batter and reso.  That gives me the D3 note and good resonance that I like.
8. I checked to ensure that the snare wire strings (old school) went into the holes (in the end plate) from the top surface and then underneath the end plate to the throw-off and end plate.

Anyhow, I'm happier now than I was a few days ago, so I thank Rick Everglade and Thin Shell for your help!

Regards, Marty Black