Your explanation makes sense to me, an uninvolved spectator with no particular knowledge of generators.
Maybe you direct the generator exhaust over the exterior of the tank? Would that be adequate heat?
My thought was to recall that my 20# propane tank has some sort of safety valve integrated into it that will clamp outflow if there is too big a surge in flow. Flow needs to be 0 for a minute or so for it to reset. But that doesn't explain your experience with the larger tank.