Chromcast should work, you may just need to do an initial setup / update when you have a WAN connection. After that initial setup you should be able to cast to it.... the only hangup might be if there is an app for the kindle.
Same might go with the fire-stick. Do an initial setup, add it the cabin network without outside access and try and screen mirror.
They also sell a HDMI adapter for the kindle fire which allows you to hardwire things.
https://www.youtube.com/watch?v=cjwnxwFDDkE