Well the answer to the question that didn't need answering...
You are allowed take your bike on the lower piste in Winterhorn.
It's not exactly advertised, but I called them to check beforehand and they didn't bat an eyelid when we turned up on a lovely sunny Wednesday recently
It's great fun, but it's not easy getting the bikes up on the lift!
I'll be happier when I can get back on the proper bike trails