Unfortunately, you'll need Google Play services to do the account check. Blackberry folks have installed Google Play Services on their devices with success. Maybe start with trying that?
I also tried installing NZB360 on the FireTV so I could integrate it with Kodi. Play services can't be installed without root, and the newer FireTV's have been locked down pretty hard by Amazon. It's pretty hard nowadays to find a NIB rootable unit.

I gave up on it. I was able to install the free version of NZB360 and it get it running, but it wasn't fully functional anyway. NZB360 doesn't format well in landscape mode, it's hard to navigate the interface since not all items highlight when selected, and as far as I can tell some options in the app simply can't be accessed with the remote.