I also sometimes use the mbasic.facebook.com site from a private Firefox tab on my iPhone, but FB has just started telling me I need to use Chrome

WTF.

But really, using a Chromium-based or Safari-based browser in private/incognito mode will not be much different as far as tracking goes.

You might also be able to install a user-agent switcher extension in Firefox. I thiiiiiink Firefox supports extensions on iOS now, right? If not, you can try an alternative browser like Duckduckgo or Orion.