I asked a similar question about this a couple of months ago and got a really good answer from /u/samsu-ditana https://www.reddit.com/r/AskHistorians/comments/9675pt/what_did_the_byzantine_empire_the_eastern_roman/

They (that is, the court and other elites) were definitely not fans. It's not that they felt that they ruled those Western regions, but they felt that the Roman title was theirs, and that Charlemagne's coronation was a bald-faced attempt to delegitimize their rule.