Bummer about the smart TV Amazon app on your LG - a friend of mine has one and she hates it (the app platform), and given that LG seems to be all in with the WebOS you'll be lucky to see any updates coming from any other devs outside of LG. FWIW, the Tivo app is actually pretty nice and a bit nicer to use than the iOS app IMHO - I haven't had time to test out my new Fire Phone with Miracast (the 32GB unlocked version I picked up on that "Fire Sale" last week - there were lots of OS and app updates); it's on wifi-only now and I'll be picking up a Cricket SIM next week to try it out on the road.
FWIW, my take is to try a Roku 3 for Amazon Prime if Apple gets the short shrift with a dedicated app or some other means. Notice here that I'm not recommending a Fire tablet or phone - I'm the same way with Android, especially non-Nexus devices. Amazon did a pretty decent job or locking down their OS fork, but, it's still Android - nothing important will go through or on it. I'm hoping for the consummate Prime video experience on an Apple TV - given what they did with their iOS app and firing a bunch of dev staff this past week, I'm hopeful...
As to Longmire, the only bad news was that directed to my mom - a huge fan, but without internet access given where she lives. Seriously, go about 100 feet to the west and there's no cell service either! So, she and I will have to figure out how to get her a fix - I do have UL data with VZW for a SIM in an iPad, but there's literally no coverage there.
Two more bits. I'll have to check out The Man in the High Castle - your religious zeal in boosting it has me intrigued! And, I made some network recommendations to a friend who was griping about the same network issues as you are here and I've not heard any negative feedback from him since - I'll check in and find out if my tweaks worked and get back to you here or via PM. Gotta. Have. Our. Fix. Cheers!