Hi there, John,
I've never actually threaded a back-plate, though there are a couple on the horizon.
I'd screw-cut using the lathe change-wheels or QC gearbox to remove most of the meat, the profile of the screw-cutting tool doesn't need to be ever so accurate. Then use the tap or taps to finish the thread to the correct profile.
That way, the initial screw-cutting establishes the axial alignment and pitch accuracy of the thread, which would probably not be achieved while wrestling with a tap as you've described. Then, the tap or taps will follow the screw-cut thread with much less effort (a smear of tapping lubricant would help) and just remove enough material to bring the thread to its correct shape.
Then carefully bore out the section that fits the mandrel register.
I see that someone else has posted while I was typing - probably beaten me to it! Hi there, Andrew!!
(Returns to arm-chair!!

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