"Periodically changing user agent" sounds pretty unique if you ask me. Especially if the extension isn't super-clever and changes the user agent for accesses that happen on the same page.
And the fingerprinter could be super clever and look for features that your purported browser isn't supposed to support... And if your browser does support them, that's a strong identifier.
Your UA will correlate with other means of fingerprinting you making you more common. Being clever can make things worse.
For example, the most common UA is from an iPhone, but the most common screen width is 1920 pixel. If you decide to make your UA an iPhone with a 1920 pixel screen, then you will be easily identified.