If this were mine, I'd consider carefully marking/photographing everything and taking the governor apart to look for any kind of clue such as broken springs, wear, sticking parts, chips/shavings, etc. I'd do it with clean tools, good lighting, and a clean indoor workspace. Maybe something obvious might show up.
I'd first try to get a feel for the actual temperature of the governor at quitting time. One of those digital BBQ food thermometer might do it. Suppose it quits at 150F. I'd bring it in the house and look for smooth operation of the shaft. Then I'd put it in the oven at perhaps 100F and check it again and then 125F and so on. Maybe something is sticking at a given temperature. Then I would see if something like WD-40 would help. If no luck, I'd proceed to take it apart.