I found a Chapman 7331 screwdriver set with four slotted tips on eBay. It's a very good price, and might be here by the end of the month. I'll wait until it arrives before tackling the typewriter again.

Progress! Of a sort.
I found the 1944 Army service manual here.
https://maritime.org/doc/typewriter/index.php
I printed out the pages about removing and reinstalling the carriage on the L. C. Smith, followed the instructions and almost got the carriage out of the machine before it hung up on the vibrator. A thin steel rod, about 2 inches long and a tenth of an inch thick fell out.
Since I couldn't get the carriage out I examined everything as best I could, then reassembled everything, wishing it was as simple and direct as the Underwood.
It still has the same problem, but since I didn't readjust the bearing retainers it's loose enough that I can rock the carriage and work past the obstruction in the middle. Theoretically, I could type with it, but I want to fix the actual problem.
But that broken length of rod makes me think there's something underneath where I still can't reach that's blocking the movement.
I could buy another machine and use this for parts, but except for the blockage in the center none of my other typewriters type as cleanly or work as well. So I'm not giving  up yet.

I did the parts swap on Sunday. The end result was a typewriter that was completely locked up.
After reviewing every possible mistake I might have made, I decided to see if I had the geared rack in backwards. 
It's completely symmetrical, so there shouldn't be a front or back, but I can look at the scars left from the previous installation and judge whether the current installation matches. Maybe there's something too subtle to see.
I had it in just like on the other carriage, so that's not the problem. I ran the loose rack over the gear on the frame to see if there was something I wasn't seeing and it ran very badly, so I went and got the original rack to compare.
The original rack is 10 teeth to the inch, while the replacement is 12 to the inch. So it's a different gear, which indicates a different font, and I would have to gut both typewriters to swap out all the internals to make it work.
My next plan is to swap the good carriage over onto the rusty typewriter and see if that works properly. If so, and I have a lot of confidence, then I'll clean and polish the rust bucket and swap as many good parts over as I can to fix it up.
I suspect my original typewriter will become my parts bin while the beater I bought for parts will become the working machine. Not what I had in mind, but I'll take what I can get.