My fridge is a bit fussy about changing from AC to propane. When I unplug from power I can hear the change over to propane right away most of the time. However early on a couple times the propane did not come on as required and the fridge gave a warning. After a bit I found if I turned the fridge off I could get the propane to work right away once I turned it back on if I selected propane and the auto.
2015 Seneca TS 5500 miles
jeep wrangler TJ
Ours is somewhat the same. I've found that if I light the cooktop for a few seconds that the fridge will start right up on propane otherwise it takes 2-4 cycles of switching it off and on for the fridge to light.
2017 Jayco Redhawk 29XK
2015 Wrangler JKU
Blue Ox Baseplate w/Avail Tow Bar & Protect-a-Tow