Amazing: Someone with knowledge of both OSX and Malayalam orthography answers my question within hours of my posting it. Thank you, Tom!
I tried your suggestions. Disabling the Apple-provided fonts didn't seem to do anything, and moving them to the trash transformed the Thiruvananthapuram page I linked above into a sea of placeholder characters. So I put them back and tried editing the preference list. I inserted Anjali Old Lipi above Malayalam Sangam MN, and I also inserted Rachana (another free font available from the Kerala government site above) above Malayalam MN. Then I restarted.
At first I thought it hadn't worked, because the body of the Thiruvananthapuram page looked exactly the same. But then I noticed that the page title, up in the top edge of the window, was displaying the characters I wanted. And when I tried copying some text and pasting it into TextEdit, it appeared in either Anjali (when I pasted into a plain text document) or Rachana (in rich text). Finally it dawned on me that I needed to adjust my Firefox preferences too, and that did the trick. Probably I wouldn't have needed to edit the preference list if I'd thought of changing my Firefox preferences earlier, but I'm glad to have learned how to make the change systemwide.
Slightly off-topic: I've read that acceptance of the 1970s reforms has been less than universal, and in the process of finding those alternate fonts I noticed that a different fork of Anjali has actually had some of the conjuncts removed. If you happen to know, I'd be curious: Just how much of a free-for-all is Malayalam spelling today?