I don't know why it didn't occur to me to try this before, but you can enable the Help -> Check for Updates menu option by launching Firefox through sudo. For example, open a terminal window and type:
$ sudo /usr/bin/firefox
Don't put it in the background with a trailing ampersand, as is probably the usual way to launch most GUI apps from the command line, or else you won't be able to type in the sudo password.
It turned out for me when I wanted to get the 2.0.0.3 update (from 2.0.0.2) that the patch wouldn't install properly, for reasons beyond my understanding, so I had to click the option to get the whole program. At least the rest was automatic -- no need to download and unpack a tarball.
Next issue: how to prevent Ubuntu's Update Manager from offering the download the updates for v1.5.0.11? Probably I have to remove it from the list of packages somehow, but I'm too tired to fiddle with that right now. Let me know, or I'll try to remember to post an update.
