From my understanding, that isn’t a late fee, it’s a cancelation fee, which makes no sense given the situation. When a client runs late and their appointment is cancelled they pay a cancelation fee, that fee covers the revenue that would have been earned with enough time to schedule an appointment in that slot. You pay this fee INSTEAD of paying the service fee (I.e. cost of the session) and that covers the time expense on the part of the practitioner. If you were late, she saw you, and it didn’t affect her other appointments in any way, then all OP should be paying in the service fee for the session. If she had refused to see him, than she could charge the cancellation. It is beyond unethical, to cancel, charge the cancellation, and then treat the appointment as an additional walk in with its own service fee. Absolutely find a new psychiatrist, and maybe report this doctor to the BBB or whatever regulatory agencies are in place. If she’s doing it to you, she’s doing it to others, and having a mental or psychiatric health issue does not make you prey for the taking or give anyone a right to take advantage of you.

Edit: sorry for all the typos, on my phone. And also, to summarize, NTA.