We performed a similar change to move our MM to a new network, and reached out to support for assistance. We changed the IP of the standby MM and created a VRRP address for the new network and assigned the new IP as the Management IP. We pointed one MD to the new MM Cluster IP and rebooted it, pointed the second MD to the new IP and rebooted it. Repeated these steps on the 2 MD's on the second cluster one at a time. We then changed the IP of the Primary MM and added it to the VRRP.
We did have issues along the way with licensing passphrase changing and an issue with some of the MDs having to be configured in disaster recovery mode because the MM IP was configured locally on the MD before it connected to the MM, which caused it not to properly change using MM.
It was pretty impactful given the issues we had and did involve reboots of the MD's. I would agree that having support or professional services involved would be ideal.
We were on version 8.7.1.4 at the time.