Fixed!
I took out the newly installed RTC (Lithium) battery, turned the unit on, and the unit reported January 1, 2000 as the date. And it let me set today's date. Of course, when I turned it off, the unit forgot the date because the RTC battery wasn't there.
So I reinserted the RTC battery, turned the unit on, it reported January 1, 2000 as the date, and it let me set today's date. And when I turned it off, it remembered the date.
Problem solved--without a firmware update, too.