How can Run Agent work in a record-triggered Flow if the async path runs as System?

I'm using Run Agent from an EmailMessage record-triggered Flow:

Incoming EmailMessage → Async Path → Run Agent

The agent works correctly in Agentforce Builder and when I debug the Flow as an admin.

When a real EmailMessage triggers the Flow, however, Run Agent fails with:

“Looks like you don't have access to this feature.”

The Flow error shows Current User: System.

My Default Workflow User is already set to an Integration User, but the async path still runs as System.

Is there a supported way to have Run Agent execute as a dedicated user with the required Agentforce permissions? Or is there another recommended pattern for this use case?