It is a good day already! I learned something new. I would have assumed that such a useful product existed, but I had never heard of Instamorph. With such a great idea, there must be some uses for it around here… 
Good point. I can buy a 5/8" to 3/8" female to male adapter, and the GoPro mount could have a 3/8" female hole tapped into it and that could be braised together.
I assume that the standard mount in a GoPro is a 1/4-20 screw (which was standard in cameras for years). If so, that same company has a 1/4-Inch Male to 3/8-Inch Female Mic Screw Adapter for just four bucks.
No, the moronic GoPro design has no 1/4" adapter. The GoPro camera is a rectangular box with flat sides. It has to fit into aplastic shell, and the bottom of the plastic shell has a pair of flat blades. That meshes with accessories with three blades. All of which are smooth, and have no way to keep from slipping and sliding unless you really crank down on the screw, in which case the brittle plastic snaps. They sell a 1/4" adapter, which is also made out of brittle plastic.
All their parts are shit.
Frankly it’s stupid not to design a standard 1/4-20 mount into the GoPro camera. But I’m curious; why are you using that camera, given the large number of cameras out there? And from previous threads, I gather that you own quite a few. I thought their niche was action shots, and there was something about them that made them useful for, say, strapping to the head of a snowboarder.
Or, you could go to the public library and use one of their 3D printers… I don’t know whether they’ll be printing in metal though; if not, regular machining would probably be better.
It was what was available at the time. I needed a small HD camera with a wi-fi interface. I’ve grown to hate the GoPro, as they flake out regularly and the wi-fi is unreliable. But I don’t have the money to buy other cameras to try out to find a better one.
Watch ebay - I just found an Al gopro to tripod mount for $7.
But really, at this point, buy an old Sinar and a used digital back - you’ll knock the socks off the toys running around.
From your problem description, metal isn’t necessary in this part. You can design up a part and have it 3d printed, even in metal.
I had a small quadrature encoder printed the other day. It costs $11 from iMaterialize in plastic.
There were options to print it out in metal, but they were over $100.
Based on the discussion here, I figured out another approach.
on eBay, I found a 5/8" female microphone to 1/4"-20 male stud adapter. And a 1/4-20 female to GoPro metal adapter like usedtobe did. I’ll connect them with either Red Locktite or plumber’s epoxy
How well do they handle being hit with sticks by hyperactive 7 year olds?