There are plenty of extensions that will let you change the UA for any site; having yet another one with such a narrow function seems to me like those "we have a smartphone app too" whose only purpose is to open the browser to their site.
But if it's your only use of such a thing, as it is for me, it's just one click away. You can also feel reasonably safe as the extension works only on Skype's web sites per Web Extension manifest.
There is an extension, that will change the UA only for Skype: https://addons.mozilla.org/pl/firefox/addon/firefox-web-skyp...