If you're concerned about possible learning issues, 5 is really too young to assess that yet. Especially if you already have access to services with the ASD diagnosis, I would suggest waiting a year or two for a neuropsych.

My son is 5. It’s really the earliest point they can do a neuropsych, but it probably won’t be completely telling yet.
You should call a few neuropsychs and get their feedback- they will tell you exactly what they will do, and one even told me they would hold off for the time being or do very limited testing.

Are the other things something a developmental pediatrician could diagnose?

If you did a full Neuropsych at age 5 or so, did you feel it was well-timed and helpful or did you wish you waited another year or so?
We have ASD1 diagnosis from 2.5 yo. I am now seeing some attention issues and also feel like we need some guidance on whether there might be learning differences and/or anxiety in the mix.

For something that's not tied to a specific public school district, and with out of network insurance coverage, where would be the best place for a neuropsych eval for a young child? Not sending to public school, so this is not for IEP purposes.