I would think a seat switch would cause the mower to die anytime the brake is off. I think the most likely problem is the reverse sensing switch. Pressing the reverse operation (ROI) switch while engaging the PTO is the best test if you have an ROI switch. If you don't have an ROI switch, you may have to check the switch itself. It should be activated by the reverse pedal and be somewhere close.
I don't think the battery is the problem because if the battery is weak, you would know it by poor starting.