You can try booking a future date and changing to a close-in date within 24 hours of the time of ticketing, as you described; if they say no, you can still cancel the ticket for no fee. I haven't heard any stories of recent success with this approach.

You can also get the United Club credit card, which waives this fee, although the annual fee is much more than $75. It would take some time for United to update your account, though, so this may not be helpful for immediate travel.

Nothing else comes to mind.