I did get back to this on Monday, and I am happy to say that I was able to finish the job.
As I suspected, to get the pushbutton control out of the housing, it has to be rotated clockwise in order to find clearance for the Neutral button. I was only able to do this after unscrewing the housing from the dash frame, and this can only be done after removing the instrument cluster because one of the three screws is hidden by the instrument cluster. The rest was not hard, once I had clear access to the assembly.
Back on the tranny end, the new cable latched up fine, and the park lock works. Regarding my access plug hack, I mistakenly said that I got it off with a 7/32" box wrench. It was really 11/32". Here's what the plug looked like when I got it out:

jt