Hi Chris, my name is Neil, and I'm a Movies & TV user like you.
I'll be happy to help you with your question.
Unfortunately, using just a Smart TV, currently there is no way for you to access and watch your TV Shows you have purchased. I keep hoping that Microsoft will make the Movies & TV app into a Smart TV app, where you can sign into your account and access your library of content, but to date, this has not happened yet.
You would at least need to hook up your TV to your computer (or cast to it wirelessly) and that way you will be able to watch your content.
You have to remember that you have bought these shows from Microsoft, so you really need a Microsoft device or app to be able to view them.
There is a way round for watching movies on a smart tv. As you have already noted, the Movies Anywhere service does not allow Tv shows. However, if you do own films too, you can use the Movies Anywhere service and sign into Amazon Prime on your smart Tv and access the Movies Anywhere films from there. But sadly, no Tv shows.
If Movies Anywhere does start to allow TV shows, this will be a way to watch them on your Smart TV without the need for any extra equipment though. So it's worth keeping your eye out.
I hope this information helps you.
Kindest regards,
Neil