I think cashmere could be a nice variation, if it's a very fine weave. Best with grosgrain, or other textured facings. The points about it being a pain to tie, and fraying easily are completely correct, so you may find it too frayed for black tie quite quickly, and then too formal to wear with your tweeds and sport coats, being sold black. If I had money to burn, I would give it a go, but since I require my clothes to last, unless they are dirt cheap, I wouldn't go with it, for pragmatic reasons.
I wouldn't do both a cashmere tie, and a white cummerbund. I wouldn't do a white cummerbund, personally, and dunt endorse it for anyone, but if you were to do it, I would keep everything else classic, so a silk bow tie. Grosgrain, again, I think, if you're going to do the white cummerbund. White satin had a particularly noticeable sheen, and so everything else would need to be staid, if I'm visualizing this properly.
I rarely find white satin to be aesthetically pleasing, at least for men's wear. Most times, if not all, I've seen it done well, it has been on a dress.
This did make me think, what does everyone think of the idea of a cotton Marcella cummerbund?
Also, for my own rig, what do you guys think of a velvet bow tie with satin facings? I go back and forth.