pb shifter won't go into reverse

I am thinking the cable is routed too close to the exhaust, and is getting overheated. I cannot think of any way the problem would be in the trans. The only thing connected to the cable inside the trans is the manual shift valve. If the trans was getting hot and making the valve bind in the valve body bore, I don't think it would work properly when cold, and/or without problems with the other gears. I had a cable in my 65 Valiant, that was very hard to push the buttons, in any gear, do to a partial meltdown of the cable, from the headers.

Is the trans and shifter totaly stock? Is there any chance either the trans or valve body is a 65?
The 64 and 65 transmissions are both cable shifted, but the valve bodies are different. The 64 used the PB's and the valve body has only 5 detent positions. The 65 used a columm shifter or floor shifter, both with cables, and had 6 detents.
Mine is a 65 trans, with a 64 valve body (reverse manual shift), and a 64 PB shifter mounted in a console between the bucket seats, in a 66 Cuda.

PS: Bob just a FYI, 64 Dart would only have a 904 trans, not a 727, from the factory, but you are correct about the cable.