Getting an Apple TV 5G (4K) or 4G for a 1080P TV?

I already own an Apple TV 4K (5G) in my main living room where I don't haven have a 4K TV (it's an 1080P LG). I choose the 4K because the 20$ difference between the 4K (5G) and the 4G is a non-sense... now, I love my Apple TV and especially how it's quick and responsive if I compare to other media player I own, such as Amazon FireStick.

I've just installed a new TV in my kid's room, which is also a 1080P TV (but brand new) and I want to install a media player on it also. I am hesitant between purchasing another Apple TV 4K (5G) or getting a refurbished 4G at discounted price from the Apple Store. Obviously between the 4K-5G (new) and the 4G (refurbished), the difference is more substantial and since it's not my "main" TV, it definitely worth considering it. However, my concern is if the 4G is same responsiveness as the 4K-5G in any circumstances such as gaming (love Asphalt 8) or any other apps. But remember both my TVs are 1080P...

Thanks! 😉