You can load the Uber Pay wallet with $100 or whatever amount of your choice. Right now they don't allow us to load the wallet with BTC, but according to the statement by Khosrowshahi, that's going to change. The transaction fee is still a concern, but not as much as the case when you need to pay everytime you book a ride.
Uber can create their own bitcoin wallet so that the users can deposit the amount with a much lower transaction fees and then use them when ever you take a ride and they can withdraw them automatically according to the charges and it will be a simple process and the issue with transaction delay and double spending can be avoided.