Rather than move the tasks, I would try to keep BI as the Publisher machine.
1. Install QVS on the new BI_3.
2. Disable the QVS service on BI.
3. Add BI_3 QVS to the cluster.
You didn't mention where the web server is currently, but I'm guessing it's on BI and you may want to move to BI_2. If not already doing so, migrate users to using a DNS alias like "qlikview" and then you can easily move it to the server of your choice.