If we are rethreading air shafts, the correct tool needed is a die, not a tap. External thread vs threaded hole. Most above-crappy ass level dies are split and somewhat adjustable with a set screw. If the specs quoted are correct you should have more than the 0.004” per side adjustment needed to get to 0.305” dia.
Or just start off threading the new section a few thou undersize, and you’ll prolly come out right on anyway.
Or you could just skim the major of the 5/16” thread a couple thou and file file file nice-a nice-a but that’s the hack way to do it.
This whole job is likely an exercise in futility anyway so hack at it...