Part of the reason the Pope used to crown Charlemagne Emperor was the nature of the ruler of the Eastern Roman Empire, for at the time it was ruled by a woman, Irene. She had first been regent for her son Constantine VI, then shared power with him and finally plotted against him, ending up with her co-conspirators blinding him, meaning she was sole ruler. It's worth noting that in 798 the Pope had affirmed that Irene was sovereign of the Roman Empire in a treaty, just two years prior to Charlemagne's coronation. But in crowning Charlemagne the Pope ruled that a woman was ineligible to be Emperor, so he merely filled the vacancy.

Moreover, relations between the two states at the time were reasonably cordial. There was an arrangement in place for the aforementioned Constantine VI to marry the daughter of Charlemagne, Rotrud, but in 788 the match broke down. Even after the coronation Irene offered in 802 to marry Charlemagne who was at this point widowed, but this didn't sit well with the court and they seized power, putting general logothete Nicephorus on the throne whilst Charlemagne's ambassadors were in Constantinople. Naturally after this union was not an option.

Notably Nicephorus refused to recognise Charlemagne as Emperor upon assuming rule. He might not have liked it but there was little he could do about it.

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Warren Treadgold, A History of Byzantine State and Society