Turns out my client was using Project Permission Mode, which there is osme documentation about stating that could've been the case here.

At the time I didn't look into it deeper as I could not see the option to change our environment to Project Permission Mode, even though I was a Project Server admin. This wrongly led me to believe it was depricated, and I kept trying to solve it using Sharepoint Permission Mode approaches.

Having access to my clients environment I could allow the needed permissions to the user created for the integration.