This ($344) is just over your $300 target but it would provide a lot of flexibility.
It is a small Win 10 PC.
https://www.newegg.ca/Product/Produc...-182-_-Product
The Shield is also good....I just felt I had more flexibility with a PC.
The Zotac or Asus would also likely work well
I live in Japan so your prices are a bit better than what I can get.
The NUC I set up (Kodi) is being used in a beach-side hotel bar and has been working well for the past 2 years.
I plan to play with a tiny Raspberry Pi as a Kodi server next - it has limits but I think there are less limitations as each new version is released.