Since the download URL is only accessible if you are signed in, and accessing the snapshot with s3 is also not possible for some reason, your only option is to make a request to the download URL with the same authentication headers you have on your personal computer. In this example we'll copy the whole download request as a cURL command from the Chrome inspector:
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Log in to Backblaze in your browser, and head to the Snapshots page.
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Open the inspector (F12) and go to the "network" tab.
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Start downloading the file (you can cancel it immediately)