From my understanding the Eastern Roman Empire considered Charlemagne to be a client king that was nominally under the banner of allegiance to the Roman Empire and considered him to be a usurper and illegitimate emperor when Pope Leo III crowned him as "Emperor of the Romans" in the year 800 AD. I was also taught that the Eastern Roman Empire would attack and try to dethrone usurpers whenever they popped up. So why didn't the Eastern Roman Empire make a large scale effort to invade Francia and dethrone Charlemagne?
They didn't really consider him an usurper. There had been two Roman emperors in the past - at one point there had even been four emperors (or, two augusti and two caesars) - so maybe Charlemagne could be the new co-emperor in the west? It actually seems like no one cared very much over in Constantinople.
Much more can always be said but I answered a similar question in the past, which may be helpful:
What was the Eastern Roman reaction to Charlemagne being crowned Roman Emperor by the pope in 800?