Ok I think I know now what the problem is: The Fire7 tablet comes with the "Silk" browser and to be honest I never used it, since I have installed Firefox and Chrome on the device and only use those two. I just tried the regular Silk browser for the first time and I automatically see the mobile version of the website, so that is all good.
I know that using Firefox and Chrome on a Fire7 is not a common use case but maybe you can still add them to your auto-detect browser list for the Fire7 device because on Chrome and Firefox I see the desktop version of the website by default.