It sounds like you already have two subscriptions that'll satisfy most of your requirements so to get around Chromecast's AP restriction you could use Chrome on your laptop/mac to browser cast Amazon Prime from your computer to your Chromecast dongle.
I was bought a Fire TV stick and it works well (getting into Mr Robot), apparently you can side load it with Kodi (not attempted yet) as well as cast to it (no haven't attempted that either). Given the cost of Sky it's probably worth the small investment just to be able to fully utilise AP conveniently but I now use my Chromecast dongle as more of an audio streamer via an AV amp because of all the audio apps it now supports. Those include SoundCloud, TuneIn, Deezer, Spotify, iplayer radio and youtube and are controlled via a cheap android tablet (as barberbob recommends).