Yup, that did it! Many, many thanks!!
I think I must have tried just about every other permutation -- I was trying to squeeze the payload into various JSON objects and the like. It never occurred to me to just toss in everything after the command as a simple payload string.
Just to spell it out for anyone else having the same issue, here is an example of an inject node that worked properly:

(Those numbers in the payload are correct for US Eastern Std Time, in case anyone is wondering).
As far as the NTP server goes, I had been assuming that what you said is correct, but the Tasmota documentation says that doing the whole TimeSTD/TimeDST dance is required for Daylight Savings Time adjustments to be applied automatically:
You must also set the TimeZone
and Daylight Saving Time policies ( TimeDST
/ TimeSTD
).
[FAQ - Tasmota]
It seems like the documentation is a little unclear in this regard -- I interpret that line as meaning that even if you do use an NTP server, you still have to apply the TimeSTD/TimeDST. If I'm wrong about this, please let me know -- I'd much rather just set an NTP server and forget about it!