Bluesky is really touchy. I would paste the username and password into notepad. Then copy it from there and paste it in, making sure not to include any spaces in front of or behind it because that will cause it to fail.

The only other way is to purchase additional codes and create a separate Workspace for each of your brands. That way, you can log in once and switch between brand Workspaces. It keeps all of your profiles, data, reports, tracking pixels, preferences, etc. separate.