Panne: Geezer here. I've built about a jillion bows with either Jayhawk prods, or prods based on their proportions and shape.
First, Jayhawk prods with holes in the center will eventually fail through the hole. That's no problem, so long as the prod is held down tightly. A set of bow-irons should do nicely, as long as you have some sort of support for the prod thru the center... say a block of wood over the lag-bolt mounting.
The draw you're getting is pretty long... 13.5 inches. I usually don't exceed 8 inches from a 3.5 inch brace, which comes to 11.5 inches. 13.5 won't necessarily kill your prod right away, but it will fatigue faster at that length, particularly if you shoot fast, in hot weather.
Stock width isn't terribly important. If your stock is wider, and the prod mounting is really solid, wider stocks, will reduce the working-length of your prod-limbs, so wider prods make for stronger bows, and greater stress for the limbs at any given draw length.
Bowstrings: You're right the steel string is an abomination, it needs to go. But it does give you a fair idea how long your Real bowstring should be. Jayhawk's steel strings were made for a fairly low brace-height, @ 2.5 inches, as I recall. I go for a slightly higher brace and shorter draw to get a proper medieval look for my bows. Certainly making the string half an inch shorter wouldn't hurt if you reduce draw-length an inch or so. As for materials, many people use Fastflite for bowstrings... it gives very high performance, but it will also reduce the working life of your prod by a considerable margin. For many years, I used 'artificial sinew' for bowstrings on Jayhawk prods. It works fine, but will stretch a bit in use, so you'll have to take it off and twiste it up once or twice. Nowadays, I use Dacron b-50 for bowstrings. It's available at a reasonable price from archery-outfitters and makes a good, strong string. A little less efficient than Fastflite, but your prod will last much longer.
Good luck with your project. The old Jayhawk prods will give you many hours of fun backyard plinking. That's what they're best for. Geezer/David Watson, New World Arbalest.