Copper requires Full Admin access to pull from the API.
Copper has controls about who can view what information. Since the API could potentially 'see' all the information, Copper requires that a user have Full Admin permissions in order to use the API.
Since Grow uses the API to display your data, the user that you use to sign into Copper from Grow needs to have Full Admin access.
Copper has an article about setting permissions and roles for users here.
If you have any questions please email firstname.lastname@example.org or chat in on the site.