I've seen lots of those on eBay. They seem to be a generic camera that's sold by many companies. Could be an option if my current action plan doesn't work out.

In the meantime I've stumbled upon a supplier of what looks like a 3D-printed tripod mount for my Sony action cam which will allow me to use the camera without the case.

If that works well then I'd just need a generic helmet strap that has a 1/4-inch tripod mount and that would work I think.

One advantage of my Sony action cam is that it has GPS built in.