Yeah, this is not a great use case for Kaiser, or any other sinc style filtering. It was included as Kaiser is another solution people will find when searching for "how to fix blurry textures". Kaiser isn't really a fix for blurry textures in of itself, at least when it comes to mip mapping. Rather it helps work around the loss of sharp detail in the mips that happens when averaging samples which helps them feel sharper without actually being sharper. Ideally you'd want to recalculate the Kaiser filter for the exact resolution being displayed, but that's way too expensive.

However I'm not really an expert when it comes to sinc style filtering, and in general avoid it (in part because Unity's implementation was broken for so long). So I hadn't thought to try allowing for overscaled values. Certainly an interesting idea. I have no idea if it'd work.