Is there a way using azcopy v10 to upload or update a blob that has a lease on it? I checked the docs, and also ran azcopy copy -h | grep lease but nothing showed up.
No, azcopy does not support upload/update a leased blob. All the supported parameters are listed here. You can raise a feature request in github like this one: custom headers.
By the way, you can try azure cli commands which supports the --lease-id parameter.