While still at the airport, I decided to just check on my laptop how much a completely new flight to Vienna would cost.
Turns out I could get a Finnair-ticketed flight the next day for $1,300. Given I had to travel overseas anyway to get a new US Visa, I jumped on it. If I'm going to have to pay money anyway, I may as well catch some of the conference and also go to a place where the US Embassy wait times are short.
Here's the irony, though: the Finnair-ticketed flights are British Airways operated. They are the same flights BA wanted to charge me $2,500 EXTRA to change to!