I had a problem identical to this with an Evinrude once. It turned out to be a bad power pack. I could put the boat in the water and it would start up every time. But if I shut it off and then tried to restart after a short time it would not fire. Let it sit for a while and it would start right up. What was happening after running a while the power pack would heat up. Then shutting it off the heat would cause component(s) or connection(s) in the power pack to swell and not work. After it cooled off the "swelling" would go down and it would start working again. This can be a very hard problem to trouble shoot. Back in my TV repair days we carried a can of freeze with us to spray a suspect component with. A can of it might be worth the investment. Amazon.com: MAX Professional 7777 Blow Off Freeze Spray Electronic Component Cooler, FR-777-777 (10 oz): Office Products
Fair Winds and Following Seas
Bill H. PTC USN Ret
Chesapeake, Va