It is possible to disable and enable the auto update both on cluster level, and on environment level in a Managed Cluster.
I see that there is a call to <Managed URI>/rest/configuration/autoupdate when this is toggled for the environment.
Is this endpoint available in the "real" REST API?
This is the internal REST API used just by the UI. It is technically possible, but you shall not call this API directly.
AFAIK the public API (Environment / Configuration) does not have this functionality (yet).
Not any public one I'm aware of. The internal REST API can be called (you need to send proper headers in the request for this to work). This is not guaranteed to work across dynatrace versions and it cannot be done in single request as you would need to authenticate first.