Opinion on when to get assessment?

My son is 2 years old and has a host of rare medical diagnoses (he has had brain surgery). He is not currently mobile or verbal, and some of his behaviour presents as things that show up in people with ASD. His paediatrician has placed him on a waitlist for an assessment. The waitlist is anywhere from 2-4 years.

I am on the fence about whether to get a private assessment for my son now, or just wait for the public waitlist I am on. He currently has a diagnoses of GDD, which I have read has some crossover and can lead to an ASD diagnoses in young children. The therapy now is the same (SLP and PT). The difference is that I would have more resources available to him if he were diagnosed, and be able to get him more therapy (the funding is worth 2 sessions per week plus a little left over for equipment).

I admit that I am scared - not of the outcome, because he will be what he will be, but of the label. I have read stories where people with ASD are not taken seriously in hospitals (something that is really important with my medically complex son), and of how it could be used against him later (life insurance, legal proceedings, jobs, bias).

I don't know what to do. On one hand if he does have ASD I want to get him all of the support he needs as early as possible, but I also don't want him to be misdiagnosed (it is possible that some of his delays are from brain injury and it could recover) and have more challenges in the future because of it or otherwise get the wrong therapy. Should I wait? Are my fears unfounded? Your opinions and sharing of information would be much appreciated. Thank you for reading.