According to the official MS documentation - https://learn.microsoft.com/en-us/azure/virtual-machines/windows/migrate-to-premium-storage-using-azure-site-recovery there will be minimal downtime while migrating to Premium storage using Azure Site Recovery and there is no mention of SAS token in the link. You may want to share the KB article you are referring to. IMO, you can try this scenario in your test environment to understand the level of impact.
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