The sort of marker I'd use depends on the exact fabric of the lower rim. If it's leather, ask for one that works on leather in a hobby store or paint store. In general I'm sure that fabric markers are fine, though. But before you're using a marker for the whole thing, I'd just put a on a stripe or a dot, wait an hour and then pour some water over it to see if it stains. If so, don't use the marker.
Just to be sure! Good luck!