Why is it asking for my CVV over and over again?

Jason, thanks for getting back to me.

Just to be clear, this is not my first purchase on my new Apple TV, I was able to buy lots of apps when it first arrived. The problem started occurring after I switched to a family plan, where my payment methods are shared with my family members.

I've been through this process on every other device I own (1 iPad, 2 iPhones, and my mac). The other devices worked just fine when I entered my CVV. This is almost certainly a bug in TVOS.

Additionally, when attempting to set my payment type to none via iTunes, the 'Done' button turns grey, and is no longer clickable.

In fact, the documentation that you linked to (Change or remove your Apple ID payment information - Apple Support) Specifically specifies says:

"If you’re using Family Sharing, and you're the family organizer, you can use these payment methods. You can’t set your payment information to None."

As another attempt to resolve my issue, I tried switching my payment information to a different credit card. After doing so, I attempted a purchase via my Apple TV. I verified that it correctly reflected the last 4 digits of my new card, but is STILL continuously asking for my CVV.