When we go camping I download whatever we want to watch from Prime onto my iPhone. As mentioned before, you can keep a program at least 72 hours after you have started watching it, but you can keep them longer if you haven't watched them yet.
I bought an adapter for the iPhone that has an HDMI cable socket.
iPhone adapter.There are cheaper ones on Amazon but a lot of reviews say they don't work. This one is from Apple and works fine for me.
You start watching the downloaded/saved program on the phone and it can be watched on the TV once the HDMI is selected. I found that if you try to start watching with the cable already connected it gives an error message, but if you start it on the phone and then plug in the cable it works great!