How can we prioritize the rancher containers running in the clusters over other containers? If the rancher containers are not running, we have no management of the system.

it is best practice to not deploy any other workloads on the rancher management server cluster. what you can try as workaround anyway is to add reservations to your containers from the rancher deployment.

This is for the pods that run on the client clusters that rancher manages, as opposed to the rancher cluster itself. In earlier versions we used to be able to change the priority of the containers via kubectl. However that now causes more issues

My bad for misunderstanding your issue. So the problem comes from resource constraints in the downstream clusters or any other areas?

Thats not an answer to your question but you can add an Authorized Endpoint to your cluster and use this endpoint when rancher is not available.