I had similar issues with the parking brake cables for my 1951 3100 although I didn't buy them from Jim Carter. My issue was that they first pair I got were too large to fit in the brackets on the frame. The second ones I bought from another vendor didn't fit either. I wound up buying a pair from LMC and they fit perfectly.
BTW, I don't think the replacements you got require that the two pieces sandwich the backing plates. I don't even think it's possible to get the one thru the backing plate. The cables from LMC bolted to the outside of the backing plate. But they don't carry them for a 1950 3100. The do have a repair kit, which is just the cable.


Kevin
1951 Chevy 3100 work truck
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1929 Ford pickup restored from the ground up. | 1929 Ford Special Coupe (First car)
Busting rust since the mid-60's
If you're smart enough to take it apart, you darn well better be smart enough to put it back together.