Hello Max!
I understand you are looking for a feedback to the above scenario. These being said, my answer will be an opinion, it doesn't mean is the right or wrong way to go.
The strategy I would follow is, before upgrading Cluster-1 to disable traffic sent to Cluster-1 (basically sent traffic only to Cluster-2) and then start upgrading Cluster-1. Test that everything is fine on Cluster-1 after the upgrade and if yes, switch the traffic to Cluster-1 and disable the traffic sent to Cluster-2 and perform upgrade on Cluster-2. After you confirm everything is fine with Cluster-2, you can distribute the traffic to both clusters.
As per your statement "We saw an unanticipated issue in our lower/test AKS environment after an upgrade, so there's always the worry that it could happen in production, regardless of what we've tested in the lower/test environment.". It would be important to understand what was the issue. If there was an issue that you could've anticipated, try to understand it and make sure you overcome it next time. I recommend you to take a look into this link ([https://learn.microsoft.com/en-us/troubleshoot/azure/azure-kubernetes/welcome-azure-kubernetes) because most common scenario that an upgrade can fail are exposed. If there was an issue caused by AKS as a platform, then it would be good to open a support case to report it, get RCA so AKS team is aware of it and fixes it.
Regarding the downside that you would need to separately maintain both clusters, it may not cover all your scenarios, but I would recommend taking in consideration Azure Kubernetes Fleet Manager. Please note this is in preview.
Reference links:
[https://learn.microsoft.com/en-us/azure/kubernetes-fleet/overview
[https://learn.microsoft.com/en-us/azure/kubernetes-fleet/quickstart-create-fleet-and-members
I hope this is helpful. If any clarification needed, let me know and I will do my best to answer.
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Thank you!