Can I use the integration user?
From what we have read, the integration user is meant to handle Salesforce users calling into Salesforce, not calling out of Salesforce. Our integration does both. It is also worth noting that the integration user license is currently quite limited, so it cannot be used with our integration.
This is because our primary sync needs to run with a user with various system permissions. Typically, this is a Salesforce administrator profile and our full permission set. Our primary sync comprises several processes wrapped in scheduled batch Apex, and are created and run by the logged-in user. As part of this, the system checks the user's permissions to make sure everything can run smoothly.
Salesforce currently doesn't let you log in as the integration user, which means you can't start this process. We're hoping that Salesforce will update this user license type in the future so we can use it, but for now, the sync should be run against a Salesforce system administrator.