Faulty starter solenoid killing car?

If the relay has the separate ground terminal it is a simple matter to run 1 wire to the transmission
Yup. Doing that today. Also correcting some other cabling they did for the reverse lights to the transmission they have sandwiched between the firewall and a metal plate. Pulled the cables, drilling a hole in the plate and installing a grommet. I'll use the same hole for the neutral safety switch.