You might try lining the inside of the shell with duct tape.  I know the idea sounds awful, but I lined the inside of my Supra to mellow it out a bit and it seemed to work.  It took on what I would call a warmer tone, while deadening it slightly.  It's woth a shot... certain cheap enough.

In my experience, a Remo Fiberskyn 3 head makes almost anything sound "woody".

 
I don't think this is more possible than getting a clarinet sound of a flute... But a flute can also be played soft and warm, just like a steel snare with thick covered heads may sound a little warmer. Of course not when you play loud. I have no other idea than your getting even a cheap, used, old wooden snare and put good heads on it if you don't like the sound of a metal snare...