Calcs are run, the short answer is go with the 1/4" nps pipe. The pressure drop in the pipe will be around 0.3 psi.
Long answer:
I thought this was going to be a simple thing to model. However, I forgot about the state change that occurs when the LPG pressure drops below about 20 psi. In true engineering form I assumed that the state change isn't occurring until the propane gets to the burner because its easier and I believe more conservative pressure drop wise. See below:
Bayou Classic BG14-210,000/hr Btu
Propane- 91,000Btu/gal
Flowrate per burner-2.3 gal/hr
Total pressure drop in piping- 0.3 psi
Picture is the model used to calculate pressure drop using the darby w. method.