The downside of making a complaint to anyone about anything though any official system is that it can drive up stress and anxiety, this is something you should decide, knowing your ability to cope.
The next thing is that your complaint sounds completely justified. As your original details are recorded for their system you should have been asked Medicare number and if you had any concession cards along wiht next of kin, DOB etc. This is particularly necessary if they state they do charge concession rates. Failure to do so may reflect upon their ability to charge Medicare for telehealth visits however should not affect you. There is a duty of care to you by all involved in that clinic.
The Health Complaints Commissioner, in my state at least, has been most helpful to me and advised me what was and was not appropriate.
If you leave the money to one side I doubt that the most important thing between client and therapist is not going to be present in the future, and that is trust.
Truth be told I'd expect the office staff are not used to the new temporary bulk-bill telehealth rate currently in effect for selected patients and have made mistakes.
Please let us know what you decide.