I would think that the push rods would be too hard for a tubing cutter. A small cutoff wheel on a Dremel or die grinder would work effectively.
I have both a 3 jaw and 4 jaw chuck available. I was using the 3 jaw last night, experimenting with grinding my own tool bits on an old pushrod. I don't have a QCTP.
yes if you can grind your own from HSS that would work.
Any means you have of cutting to within 1/8th of the mark is better than parting on the lathe. I have a tubing cutter like this and have used it on steel tubing. With good lube on the cut it works. Since the end will be faced is the beveling a problem?
RIDGID 3/16 in. to 1-1/8 in. Model 15 Screw Feed Copper Pipe & Aluminum Tubing Cutter w/ Fold Away Reamer + Spare Wheel 32920 - The Home Depot
Built on the trust of the trades, you need tools designed for experts and built to last. Experience the durability of the RIDGID 15 Screw Feed Tubing and Conduit Cutter (catalog number 32920) with 3/16"-1-1/8"www.homedepot.com
I'd start looking for a QCTP and some cheap indexable tooling. Lantern toolposts are a PITA.