There are several accounts about the coronation, most notably from the papal perspective in the liber pontificalis, and some frankish sources (i.e. Annales Laureshamenses). There are also sources from the Byzantine Empire. The sources differ somewhat in how the coronation took place.

The papal view stresses the papal influence over the whole ceremony. It serves to illustrate the fact the the crown was received by the church, putting the latter over the former.

The frankish sources paint Karl as the main actor, but he is not portrayed as actively pursuing the crown. However, not wanting to become emperor is a trope that is known from late antiquity. It serves to illustrate the humbleness of the would-be emperor and that it was his fate to become emperor.

Most modern historians agree on that the coronation could not have been planned and enacted without the consent of Karl. The implications were so far-reaching that the Pope - who was utterly dependent on Karl for protection against the nobility of Rome - would not have attempted to do this without or against his will.

The last book I read about that topic was "Karl der Große Gewalt und Glaube" by Johannes Fried. It contains an extensive part about the coronation and how it is presented in the sources.

P.S. Karl is the german name for Charlemagne, as you might have guessed.