Yes — when a Business Manager (BM) is added to an Upwork chat room, Upwork displays a system message in the chat: "User A added User B to the room." This notification is visible to everyone in the chat, including the client.
Does this cause any issues?
In practice, clients rarely pay attention to these system messages. They are a standard part of Upwork's chat interface and are generally understood as normal team management activity. Clients typically focus on the conversation content, not on who was added or removed from the room.
What the client will see
A system message in the chat thread noting that a team member was added
The Business Manager's name and/or profile in the participant list
What the client will NOT see
Any explanation of the added user's role or permissions
Any GigRadar-related context — it appears as a standard Upwork user addition
Summary
Adding a BM to an Upwork chat room is safe to do without concern. The client will see the system notification, but this is routine and causes no issues in the vast majority of cases.