Thanks, Ann. As it happens, I'd attempted a washing of the hat a few hours earlier than your response. I used a mild shampoo (no conditioner). I "slooshed" and "sloshed" the hat clean, using lots of fresh water. I used cool tap water through the whole process.

To dry, I laid out the hat on a soft folded towel, as if it were on my head, and let the majority of the water be wicked from the hat by the towel. After a couple hours, when the majority of the water was absorbed by the towel, I then put the hat on my head while it was damp (but no longer drenched), to ensure the headband still fit. So far, so good.

The lay of the hat may not be quite what a complete "jocking" would yield, but I will say the hat came clean, stayed together, and I didn't destroy my $80 hat in the process.