Currently there is no API for those metrics. The only way is to relay on Dynatrace Misson Control to get proactive alerts if something goes wrong with the cluster.
I would be very interested in options to get more data about the dynatrace managed cluster health as well.
For now one of recommendation is to install OneAgent on Dynatrace Managed cluster but it is recommended to use for monitoring separate instance. I've seen information that self monitoring is on the list but no info about date 🙂
I would not recommend that. A self Monitoring Agent is already running on managed cluster nodes to report data to mission control.
If you overwrite it with another OneAgent and send the data to one of you environments, Mission Control will no longer get that data.