Desperately trying to get from Palermo to Lipari on Sunday

We arrive on Sunday morning in Palermo and are joining friends in Lipari.

I need some help with options:

-direct hydrofoil is fully booked,

-tried to switch to Milazzo (thank you vagabonda),

-car rental is closed on Sun in Milazzo on Sunday, so that is not an option (Messina idem, closed Sun pm),

-no direct bus,

-trains run late,

-any other options? Uber? Taxi? Would that be very expensive?

Thx for your help!