Alright, I'm an idiot. I didn't notice that there was a place on the Download Options page for me to specify the URL for the downloads page. Adding index.php/ before downloads did the trick. Thanks for the help.
By the way, the comment below the downloads page URL text control says that it should be specified without a trailing slash. I didn't touch it until now and it had a trailing slash. It seems to work fine with or without one.
Thanks,
David