@EnterpriseArchitect As my colleague @SadiqhAhmed-MSFT mentioned , Cross subscription back up and restore isn't supported scenario. The below process is a workaround that I figured which doesn't include Azure Backup and I have not have tested it. Kindly check it if it helps you.
1)Take a snapshot of the disk by following the steps in the document. If you are going to use the snapshot to create a new VM, we recommend that you cleanly shut down the VM before taking a snapshot, to clear out any processes that are in progress.
2) Copy the snapshot to different subscription by following the steps in this document.
3) Create the VM from the snapshot as mentioned in this document.
Also, to support cross subscription restore, I would recommend you to navigate here and share your feedback or suggestions directly with the responsible Azure feature team and clicking the vote button of your suggestion to raise visibility and priority on it.