We want to be able to accept deposits after confirming an booking (and/or receive/invoice a portion of the full amount owing).

YESSSSSSSSSSS - this is a must. This feature stops us from switching to your system.
We are a hair salon that takes deposits for bookings made in the salon, in person. So we need a way to record that payment in the system so we know how much is outstanding and its connected to the services.

We also need this function. We need to accept deposits in the salon. Not only on the site.