How fast are the pressures changing when you open and close the valve (if that's all you need to do to cycle the third burner off and on)?
The 2 "obvious" possibilities are 1/ that the cylinder is not getting enough heat input from its surroundings to maintain the take-off rate for 3 burners, or 2/ that there is a restriction in the system that will not allow the take-off rate for 3 burners.
Possibility 1/ will show a fairly slow response to switching the 3rd burner: probably minutes, maybe even tens of minutes, for the pressure to stabilize at the new value. Possibility 2/ will show an immediate response: within a couple of seconds.
If you don't know from the testing you've already done, it's worth spending a few minutes checking to see which of the possible explanations seems to apply in your case. You don't even need the forge up to temperature for this.
The restriction could be a safety restrictor in the Propane cylinder, intended to limit the flow, or a too-small regulator, or something else. First thing to do is see whether there's a restriction, then, if there is, to track it down and eliminate it.
If it's "just" ambient temperature, a bigger cylinder, manifolding 2 cylinders together, or increasing the heat transfer to the cylinder are potential fixes.