The Admin Key feature does not support Forge apps today as it is restricted to org/site admin users only in Premium+ tenants. We've been receiving a growing number of requests to support Forge apps in addition to org/site admin users, as seen in the comments of this blogpost: https://community.atlassian.com/forums/Confluence-Cloud-Admins-articles/Introducing-admin-key-support-for-REST-APIs/ba-p/2996259?utm_source=atlcomm&utm_medium=email&utm_campaign=mentions_comment&utm_content=topic.