@raysinboise
The problem is the Handbrake preset you are using.
The AppleTV1080p60 transcodes video to H264, 60fps, High@4.2. This is supported by the AppleTV v4 (tech specs), but not by any version of the Amazon Fire TV stick/box/tv.
The Amazon Fire TV Stick supports H264, 30fps, High@4.0. See AFTV Specs
Therefore, PMS is most likely transcoding your videos when you are using the AFTV stick. Video transcoding is very CPU intensive. If the CPU in your Drobo is not powerful enough you will see buffering, pausing, or error messages like what you are experiencing.
Most DVDs & Blu-rays have video at 24 fps (at least in US). The video on most Blu-rays is H264, High@4.1 (some have VC1 video). Increasing the frame rate from 24fps to 60fps doesn’t buy you anything. Neither does increasing the profile and level. All it really does is buy you compatibility headaches.
For Blu-ray discs (1080p), suggest you use AppleTV 1080p30 preset, change framerate to “Same as source” instead of 30 FPS, and Constant Quality to between 20 - 22.
For DVDs (480i/p), suggest using the AppleTV 3 preset, change framerate to “Same as source” and Constant Quality to 18 - 20.
If you are loading the videos into iTunes, then have Handbrake create files with the m4v extension. If you are not using iTunes, then choose the mp4 extension. There is no difference between the two. You can rename .m4v files to .mp4 and visa-versa. iTunes likes movies to end with m4v. If you load movie_name.mp4 into iTunes, it will still work, but iTunes will ignore chapter markers. Plex does not seem to care. It has never given me problems with movies ending in m4v.
Note: If you have high framerate media, above 30fps, then choose 30fps instead as “same as source.”
You also need to pay attention to audio compatibility. Transcoding audio is not nearly as CPU intensive as transcoding video, but it does hit the CPU.
For maximum compatibility, have one audio track with AAC codec, 320kbps max (AppleTV limit), and Dolby Pro Logic II mixdown (or mono/stereo if source is mono/stereo instead of 5.1).
For the second audio track, choose AC3 passthrough if the movie has a native AC3 5.1/2.0/etc audio track. If it does not, then choose AC3 codec, 640kbps, 5.1 mixdown (or mono/stereo if AC3 1.0/2.0 source).
The resulting file will play nicely with iTunes, Plex, AppleTV gen 4, and Amazon Fire TVs. You can also load it on recent iPhones, iPads, and Android phones & tablets.
You’ll have a movie with H264 High@4.0 video at 30fps or less, AAC audio for non-surround sound devices, and AC3 audio for home theater & devices that support AC3 audio.