Changed an integration user from system Admin to API-only - ERROR: Failed: API security token required

For an integration between two Salesforce systems we have been using a dedicated user with System Admin profile, but we maxed out our 10 free NPSP users, so I changed that user to Minimum Access - API Only Integrations profile. 

Now I'm seeing these errors in my Login History: "Failed: API security token required" 

I think there's supposed to be a way to exempt a user from having to provide a security token, but I'm not sure. 

Any advice is appreciated.