The "making it catch up" is the tricky part. You need an initial backup for that. xtrabackup can take that backup "hot" without blocking read/writes. mysqldumper will block writes for whatever time that initial backup takes, for 2TB of data that's going to be hours.

Once you have that initial back up you can set your replica and make it catch up , then you switch. I choose to take the few seconds of downtime doing the switch because for my use case that was acceptable.