Glad you found us, Darne! Welcome aboard.
First, let me explain about this thick side and thin side stuff. On the HIGH-wall, there is a flared end configuration, which is milled down from the full thickness (thick side) configuration. In the low-wall, the one that has NO flares is the thinnest yet, since the flare side is the same as the flare side (thin side) high-wall with the sides also lowered and the smooth sided low-wall is what's left of that when the flares are milled off! If you are confused by all of this, just take my word for it that the Winder Musket low-wall (also called 3rd Model Winder Musket) is the strongest of the bunch and will take anything you can safely put in ANY original vintage low-wall.
I would say that the .22 K Hornet should be OK, but if you would feel better about going back to rimfire, that would certainly be a step toward additional safety. I would be somewhat hesitant to RE-alter that breech block though, and you would probably have to come up with another extractor. If it were me, I'd go with the restocking job and see how it shot, then if it STILL isn't up to the standards you need put a high quality .22 rf bbl on it and get another block and extractor. The theory of fore ends for 'walls is a bit more complicated than I want to put in this already overlong post so I'll get back to you on that. 8)
BTW, the first 'wall I touched, and now own, is a 3rd model WM that I have fitted with a PG sporter stock w/swiss butt plate, a single set trigger, a Shilen barrel in Win 52 bull profile and an 8X Lyman TS Jr, and I love it!
Again, welcome to the board and good luck with your low-wall, but watch out, they're addictive!
Green Frog