>> "If I call TA_CheckAndSavePKey() with a null string and TA_SYSTEM from the installer (which will have admin permissions) will that *also* set up the necessary fields for TA_UseTrial with TA_SYSTEM to work from a non-elevated instance of the program?"
No. The activation and trial systems are completely separate.
>> "Or is there a way to call TA_UseTrial without actually starting the trial period? "
No.
>> "Of course, I can always fall back to TA_USER instead of TA_SYSTEM, but prefer not to."
That will work. Or, on Windows, 99.99% you can call TA_UseTrial / TA_CheckAndSavePKey() with TA_SYSTEM without admin permission. You only need admin permission when a customer has seriously messed up their computer.
You will *always* need admin permission for TA_SYSTEM on Unix (macOS, Linux, BSD, etc.)