Sounds like a fun idea. This could still be done in the absence of a basketball tournament.
You could do this using 64 NWS offices and have a “round” every couple of days or once a week or whatever.
The offices could be seeded based on climatology rather than something that’s unrelated to surface temperatures.
The predictions could follow the same rules as the basketball brackets (you have to completely fill out your bracket before the beginning of Round 1). Then you’ll have the challenge of picking which of two offices will be hotter weeks from now that are likely to be very close to the same temperature. We know that Norman, OK will be much hotter than Anchorage, AK; but they won’t meet up in the later rounds. Anchorage will be long gone. Your Championship will be more like Phoenix vs. Miami, and you’re trying to pick the winner a month in advance.
The #1 vs. #16 seeds will be obvious just like in basketball. Everyone’s bracket will start falling apart during the Sweet Sixteen, just like in basketball.