i have no idea bout the fire tv (stick) cause its close sourced and you cant access the bootloader/system software to fix that, and i dont own one to figure it out myself. But what tv are you using, many tvs have advanced hdmi settings where you can disable remote control using hdmi and thereby only enable a/v using hdmi. I dont know if your television set has such a setting but my lg tv has something like that, i have disabled this to listen to music using my stereo from the blu ray player (over upnp/dlna) without the tv switched on...
have a look at your tv user guid for example, or thell us the tv model so we could check.
If there is no way doing that by tv just unplug your tv or use a hdmi switch with an remote control between the fire box and the tv to workaround...
i wish you much suggest on this issue, i can imagine that this is a pain in the ass.
By the way dose your fire stick has a separated power supply, if so you could also switch this one on and off wireless...
regards treaki