The linked SO answer worked for me conceptually, but it didn't indicate the appropriate fileExtension. My config had to instead indicate the following. (Note, for any who have this question, by default my azure website didn't have a web.config initially, so I had to add the following to a text file, save as web.config, and FTP to my webroot.
<?xml version="1.0" encoding="UTF-8"?>
<configuration>
<system.webServer>
<staticContent>
<remove fileExtension=".mp4" />
<mimeMap fileExtension=".mp4" mimeType="video/mp4" />
<clientCache cacheControlMode="UseMaxAge" cacheControlMaxAge="7.00:00:00" />
</staticContent>
</system.webServer>
</configuration>
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