Thank you for reaching out.
I understand that you have a three-node cluster with two nodes running on-premises and one node running on Azure. In case of a complete shutdown at on-premises, you can move the resources to the Azure node. You want to know how to bring it up on the Azure node to avoid issues after failover to Azure or even after failback resources from Azure to on-premises once the site is up.
I think Using Azure Traffic Manager in this scenario will be a good choice.
Azure Traffic Manager is a DNS-based traffic load balancer. This service allows you to distribute traffic to your public facing applications across the global Azure regions. Traffic Manager also provides your public endpoints with high availability and quick responsiveness.
A very similar site recovery scenario Azure <-> On-prem is described in this article here
When Azure Traffic Manager detects that the Primary endpoint is no longer healthy, it automatically uses the Failover endpoint in the DNS response and users connect to the application recovered on Azure.
Hope this helps! Please let me know if you have any questions. Thank you!
Please "Accept the answer" if the information helped you. This will help us and others in the community as well.