HA Connection lost after change Management IP address

I would have to think and read about everything that needs to be updated and how quickly it must be updated in order to prevent an outage when changing the management IP address.

Basically all F5's in the cluster would need to be updated at the "same" time. You would also need to make sure the DSN entries for the F5's are updated so that when the F5's do their DNS lookups they get back the new addresses.

You may want to open a case with F5 support to see the best way to do this.

However, do you really have an outage? I would think that one of the F5's would keep responding.