Is it absolutely necessary for the therad to run right up to the underside of the head?
Often, the hole has a chamfer, or there is enough thread engagement for an undercut on the screw not to damage security.
Ideally, as long as you can get 1D (one diameter) of thread engagement, sufficient clamp load can be exerted.
You are cutting the thread with a die; the original screw was probably mass produced on a machine specifically designed to produce the screw in its entirety, by the thousand. (Bar in, finish machined screw out)
A Cri Dan screw cutting machine would produce a perfect 1″ BSW thread, 8 or 10 inches long in under a minute, in several passes! So 7/32 BSF screws would be even quicker.
At the Bell and Howell cine camera factory in Gloucestershire, the machines there would produce a completely machined small screw in less than a minute!.
Probably you won’t have a problem.
Howard