It could possible be a lazy regulator at the tank that is not reacting to the very slow and low flow of the pilot lighting for the frig, but once there is a larger draw on it by the stove, it opens up and functions normally after that.
I have seen this happen on compressed air regulators and also on acetylene regulators, where I would have to turn them up to a higher pressure and then back down to where I wanted them if they had been sitting a while, or put a big draw on them for a bit to get them to open.
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