Check to see what iOS version your HA supports, and make sure that the new AppleTV that you buy is on that iOS version or later. iPhones and iPads and iPods are on iOS, but MacBooks are not on iOS but instead are on MacOS so MacBooks are not compatible with MFI HAs. I think AppleTV is on tvOS, no? So I’m not sure if tvOS supports MFI HAs or not, but probably not. I thought I read something about Apple trying to port iOS to AppleTV but I don’t know what AppleTV model has iOS in particular.
But since you already have your MFI HA, just try to pair it with the AppleTV in the store first to see if it works before you buy.
Having standard BT alone on AppleTV or MacBooks is not good enough for the MFI HAs. It has to be the proprietary low energy BT Apple use on their iOS to work with the MFI HAs.