A cognac alligator strap would be a nice colour choice, thought part of me thinks that an Oyster always looks best on its Oyster bracelet. If you do go for a strap, I think you're on the right track. Here's an example I just found on Chrono24.
Two other things that might work: first, maybe you could pick up a used or NOS, or even "reproduction" stainless Oyster bracelet, and go that route? Maybe the gold bezel would look lost, but I think it might be workable. Alternatively, a bit of lateral thinking. Paint the gold centre links with clear nail varnish. It might be a bit fiddly to stop them sticking together, but if you do it patiently one at a time, it will make an invisible barrier between gold and BB, and if it doesn't work out, is easy enough to clean with a little paintbrush and some acetone - no harm done.