The best source I have on me right now is that of the crusaders from France as they visited Constantinople on their way to Jerusalem during the First Crusade, where they heavily refer to the Byzantine state as "Romania", or more literally Land of the Romans.

Count Stephen of Blois, a minor leader of the First Crusade from his letters back home, refers to the Byzantines as Romania:

You have certainly heard that after the capture of the city of Nicaea where we fought a great battle with the Turks, and by God's aid conquered them. Next we conquered for the Emperor of all Romania

There are also many instances of the Byzantines being referred to as "Empire of the Greeks" to delegitimize their claim to the Roman Empire, as far as I know this was almost always intended as an insult to the empire and used mainly by Italians and the Papacy to reassert their claim of the Holy Roman Emperor being the legitimate continuation to the Roman Empire.

By and large, Western Europe did refer to the Empire in Constantinople as the Roman Empire, or Romania for short, despite insistence that the Holy Roman Emperor was the legitimate claim by the Papacy.

Its a misconception that Western Europe saw the Holy Roman Empire as the only empire. It was not a situation like China and Taiwan, today were both claim to be the legitimate rulers over the same territory. There were areas of disputed sovereignty but fundamentally they did not claim to be the same state. The setting up off the Latin Empire, after the fourth Crusade, which was called the Empire of Romania with no political problems.

In commutations between Frankish states and the Eastern empire what they called each other varied with there current relationship. When the Italian maritime republics were allied they address the Emperor with his correct title. When they were hostile the Emperor of the Greeks. Even communications between the two Empires varied dependent on the state of relations.