So the apple tv hdmi connection via the scart converter works, the fire tv one not?
Did you try to set the resolution of the fire tv to 720p (using a minitor or other tv set with hdmi input)?
I used an active hdmi to s-video converter from ligawo to connect a fire tv stick to a Loewe CRT.
So there is a solution for you for sure.
Ralf
Living Room: Beosystem 4, Beolab 7-2 (Center), Beolab 9 (Fronts), Beolab 8000 (Rears), Screen: Sony KD-85XH9096
Home Cinema: Beosystem 4, Beolab 7-4 (Center), Beolab 9 (Fronts), Beolab 4000 (Rears). Projector: Sony VPL-HW55
Home Office: BeoVision 11-46, 4 x Beolab 3, BeoCenter 9300 (BLC 1611/ML/NL)
Bedroom: Sony KD-65XH9077, Beosound Essence MK II with Beolab 6002 and Beolab 11 (all white, wall-mounted)
Dart Room: BeoPlay V1-40 on floor stand
Guest Room: BeoVision 7-40 MK IV, Beolab 7-2, BeoSound Ouverture