Re: scoping a Browning challenger III
Is the pistol already drilled and tapped? If so, somebody's already making a mount. I'm not real familiar with your pistol, but it appears that the barrel, assuming the bull barreled version, is without taper, and slab sided.
This being said, or even if you have the lighter barrel, any competent machinist can take a chunk of blank 'Weaver' base stock ( See Brownells), and contour the bottom to fit, then drill and tap the barrel.
Another option might be to use a mount for a 1911 type pistol, one of those that puts a plate under the grip scale, secured by the grip screws, and cantilevers the mount over the slide; the plate would need to be re-drilled, and contoured for the Browning frame, but would not alter the pistol in any way. Seems like Pachmayr sold them, at one time.
Hope this helps a bit.
Don't start no s**t and there won't be none, Terry