> the Medieval Empire, that is what is commonly referred to as the HRE,
Sure, the HRE name is three lies for the price of one, but its at least specifically understood in its referent; “the Medieval Empire” is...not. Byzantium, most notably, being a thing.
I'm using that terminology because, unless stated otherwise, in historical European documentation Empire used by itself usually means the HRE.
Furthermore including the Eastern Roman Empire as part of European Medieval history is dubious, or at least novel (and I'd wager the collapse of the Ottoman Empire, the Cold war and the EU have something to do with that shift in perspective).
The European title precedence tradition I'm referring to, and from which the translation of 天皇 as Emperor is affiliated to, was solidified in the context of Late Medieval/Renaissance diplomacy [1], in which the Eastern Roman Empire was neither considered as part of the Western world (it was quite literally the East), not relevant as it had become a rump state.
[1]: If you're so inclined, read about the councils of Constance and Basel-Florence-Ferrara. They involved a lot of international politics, petty quarrels on precedence and as a bonus, the almost-but-not-quite end of the Great Schism. My favourite tidbit is the Spanish and Swedish bishops arguing over who among them were the true Goths.
Sure, the HRE name is three lies for the price of one, but its at least specifically understood in its referent; “the Medieval Empire” is...not. Byzantium, most notably, being a thing.