Obviously doesn't help when using an app, but what about an extension that changes your user agent every single time you send a request to a website? Does such an extension even work, or exist?
Even if it did, it will be useless because fingerprinting (of various kinds - whether TCP or TLS stack, doesn't even have to be JS-based) will still tell which browser you're running on.
Worse, it may actually make you more identifiable because a Chrome on Linux user agent with the TCP fingerprint of Windows and TLS fingerprint of Firefox is going to stick out like a sore thumb, even more than a rare-but-at-least-consistent fingerprint with no trickery.
I'm not sure TCP fingerprints are very reliable because a lot of ISPs and Mobile Carriers are doing proxying at the application level for HTTP/HTTPS ports so even if you are on an iPhone you might end up having a linux TCP fingerprint. This site allows you test your TCP fingerprint: http://witch.valdikss.org.ru
