John Deere has been selling special starting kits for garden tractor type machines. Likely would solve your problem as well.
High power needing stuff like a starter solenoid or a fuel cut off solenoid are being feed their power through many safety switches and electrical connections. When the power finally gets to say the starter solenoid there are not enough amps to pull in the solenoid.
The relay kit becomes the equivalent of jumping the starter without loosing all the safety features. The relay coil can operate on a poor electrical supply but then send full battery voltage to the required circuit.
This is a home made one on a Cub Cadet. An automotive headlight relay and 10 minutes of work.
Dave
M7040