Yay! I can use Firebird to access the Internet.
I then copy-and-pasted our local proxy response, and found that Firebird was attempting NTLM authentication, not Basic authentication (as the other browsers I use do, I believe).
This made me guess that maybe the domain name was too much: I had previously entered my username as "domain/user" into the dialog box, because that's what my other browsers need as authentication data. However, when I entered simply "pne" (my local username) as the proxy username, it let me through.
So now I can also use Firebird. Let's see whether I like it.
Edit: Apparently, it entered my username as "pne!" since that is what I put in the login box (in order to set the "never expire login cookie" flag) -- this caused "Client error: no username sent" when I posted this entry. But that's minor.
Edit 2: Yay! And it can even display a navigation menu if the page uses the <link> tag. Nifty. (This was one reason why I didn't upgrade from Mozilla 0.9.8, since they removed that feature from 1.0 IIRC.) And it's interesting to see what sites support navigation... for example, LiveJournal does, and Fastmail.fm.