This was a common problem on many MTD built mowers. The power to the starter relay runs thru all the safety switches before it goes to the starter relay. One of the switches likely has high resistance when warm and is reducing the voltage to the starter relay. The cure recommended by MTD was to add a relay to the system. Here's the wiring diagram from MTD for adding the relay.
Starter relay wiring
That's great if his is wired that way with a solenoid on the starter - some are; some aren't. Josie, this is going to be hard to diagnose correctly until you get the correct schematic for his exact model/serial. Hope that the dealer rep sorts it out without you having to.
- Jay