I've been going over your setup and the one significant difference I see is that you have the Dual tuner tablo. You also have a different router and hard drive, but I wonder if there is something about the dual tuner version that works better with the current firmware.
Do you have the quad tuner tablo to try? Or have you also tried that one and found none of the live tv buffering issues, and reconnection issues when leaving a playback using the fire tv stick?
On the router, which security encryption do you use? WEP or WPA? I think I noticed mine is set to WPA. I don't know if this has any significance but thought I'd ask you.