Please tell me if I am wrong, however: I believe that, since my machine has english (20 thread per inch) leadscrews, I need to have the native units set to inches, and then let Mach3 do the necessary math to convert metric coordinates as necessary for the machine?
You are wrong. 20 turns per inch = 1.27 turns/mm Mach3 doesn't care what pitch your screws are. You just tell it how many steps to take to move a given unit, regardless if that unit is inches or mm's.

The metric/english setting doesn't seem to persist as part of the profile. Am I wrong about this?

Mach3 will always start in it's Native Units.

Clone your existing profile, and create a separate shortcut for the metric profile. If you don't know how to use the /p switch in the shortcut, see the Mach3 manual.

Set your native units to metric, and divide your steps/unit by 25.4. You'll need to convert your velocity and acceleration values to metric as well.