According to @megahorst
"Kindle Fire 6 tablet and the Fire Stick.
The App starts and streaming works on both devices."
Yes! Did a short test yesterday and it worked fine. Sideloading was no problem with Apps2Fire (thanks @megahorst for the hint!).
Profile "No Transcoding" had no video, just audio, but High Quality FFMPEG worked fine. Usability with Fire TV's remote is not so good, e.g. pausing and skipping in live tv is not possible, but I will give amazon's "Fire TV Remote App" on my tablet a try and hope that I can swipe and tap with it.
Will report if it works or maybe @megahorst already tried it?