I've been (slowly) working through a Colt Baby Lightning, the 22 cal version.
I'd suggest that the first course of action be to buy a screw pitch gage, the largest one you can find, as it will have the most pitches to compare against.
Use a splinter of wood,screwed into the hole by hand, to pick up the thread size (diameter) and pitch (number of threads per inch). My money is on them being in a diameter/pitch combination that you will not find in a store anywhere. Nothing new, gun makers have been using proprietary thread pitches and diameters since about the time guns were getting made.
Once you know the pitch and diameter, measure the head diameter, depth of the head recess, the length of the shank, and the diameter of the shank (esp. if it goes through a hole larger than the screw thread), then sit yourself down and draw a sketch with all that info on it.
Once that is done, you have a pretty decent chance that you can convince someone with a lathe to make it.
If your lathe guy only threads stuff with taps and dies, find a better one. You need someone that can thread cut some oddball sizes.
Best of luck!
Cheers
Trev