Ha ha nice, that's awesome!To clarify for @sideshowuniqueuser, you install the tvOS beta profile on your iPhone. Go to betaprofiles.com and scroll down to Block OTA Update. That causes the phone to only look for tvOS updates that are compatible with it, and of course there are never any tvOS updates compatible with the iPhone. Therefore it never thinks there's an update and you never have to deal with those nag messages. This trick has been around for at least a few years now.
The reverse used to be possible too (install iPhone beta profile on Apple TV to block those update nags), but tvOS 13 refused to allow installation of a non-tvOS profile. I'm frankly surprised that iOS doesn't yet block installation of the tvOS profile.