Old biscuit good.
I had a solenoid valve go bad in mine. It was doing similar while it was in the process of going bad. The burner was not getting enough propane, there would be a little propane at first, enough to light a flame, but then would run out of propane bc there was not enough pressure going to the burner, so it would shut off, then I guess a little pressure did build up in the system by the time it lit again and it could burn for a second but just go out when starved for propane.(Im not sure if this was what was really going on but was just my theory).
I replaced the valve.
I guess if the pressure feeding the furnaces is too low this could happen too. Unlikely as I am assuming all your other propane things work just fine?
How old is all this?
Regulators do go bad after a while. 15 years is a long time for a regulator.