PGBouncer Pause Waits for Remote Connection to Close
At least with pool_mode=session, it looks like a 'PAUSE' command will succeed only if the other client(s) have not yet issued any queries in their sessions. If the other client(s) have already issued any queries, the 'PAUSE;' command hangs as you describe.

This basically makes sense from PgBouncer's point of view, as it doesn't need to give a real connection to a client before the client issues any queries, but once it does a 'PAUSE' must wait for that client to finish its session.