3. Charlemagne claimed supremacy over all Christian kingdoms, even those he had no secular power over (e.g. Mercia), which is once again reminiscent of the caliph more than the Byzantine.
4. Why would the rising power of the Francian Empire aspire to emulate the old byzantine empire that, in large part by the hands of the Caliphat, had long passed the zenith of their power. There is some value in "reconnecting with an ancient heritage" for propaganda. But if you wanted to build a resilient and successful empire in the 9. century, who would you emulate? The falling star of the ancient Byzantines or the Caliphat spreading from Iberia to India?

See: babka