That is exactly my point. SIC should work out of the box, no additional hustle, if everything is done correctly.
You do not need to re-establish SIC.
On old management you should add a dummy object as a secondary MGMT server, with IP address of the new MGMT server. After pushing policies on all FWs, shut down the old MGMT server and check SIC from the new one. It should work. If it does not, something in the middle is interfering.
All that under assumption you did not disable control connections in implied rules. If you did, on old MGMT you should also add an early explicit rule allowing control connections to/from FW & new MGMT IP.