If you want to just do it the "easy" way and have the minimum number of PSets, completely bespoke for this integration:
- Make a user with the Minimum Access - API Only Integration Profile.
- Create a new Permission set that has the "Salesforce API Integration" license. There is a very similarly named license that will not work. Be sure you are using this one. Add all the permissions you need in this pset.
- If you are working with packages like Salesforce CPQ, you may need to make additional PSets that have Permission Set Licenses. Those permission sets cannot apply App or System Permissions. **You cannot use any Permission Set that requires a different User License, such as "Salesforce."
- Assign this/these Pset(s) to the user
- Assign the Integration PSL to that user