Ok, I think I have found the answer / solution by myself.
As I have stated out I had a lot of errors and warnings in the failover cluster log which you can access through event manager.
The next hint was this old technet post
where Erland stated out that NT Authority\network service did not had sufficient rights.
I checked the properties of my cluster object from Failover cluster manager and the network service only had Read Access.
With the powershell cmd Grant-ClusterAccess "NT Authority\Network Service" -Full
the high CPU usage turned from clussvc to the WMI Provider. Putting both nodes then in maintenance mode for a few minutes and then let Health Monitor resume its work did the trick.
No high CPU usage anymore.