Why did Charlemagne divorce the Lombard Princess Desiderata?

Scholars today aren't even sure that Charlemagne married Desiderata, although marriage was seriously discussed sometime around 770. The Royal Frankish Annals, which seem to have been first compiled in the Frankish court around 790, recorded that Charlemagne's mom went to Italy, and a slightly later chronicle (Annales Mosellani, ca. 797) report that she met with the Lombard king to mediate Lombard relations with the pope and "led the daughter Desiderata to Francia."

But the pope didn't like this part of the plan. In 770 or 771, Pope Stephen III (r. 768-772) wrote to Charlemagne and his brother Carloman. It had come to his attention:

The daughter of Desiderius is to be joined in matrimony to one of you brothers, and if indeed this is so, then it is certainly an entry of the devil—not a joining of matrimony but rather seeming a consort of gross iniquity, since we have condemned many, as the stories of holy scripture have taught us, for departing from the mandate of God and to have decayed into great dissolution through a wicked bond with an alien nation.

Whoa. Why such ire? Well, an alliance between Charlemagne's family and the Lombards would seriously disrupt the balance of power in Italy. Every time the Lombards became to aggressive to the pope and his lands, the pope could call upon the Franks to hit them on their flank. But if the Lombards and Franks were allied by marriage, the pope was without a defender. As almost a side note, the pope notes that both brothers already had Frankish wives—"and certainly it is not permitted that you should dismiss these so that you can take other wives or mix in consanguinity with outside nations."

These sources leave historians with two problems: (1) Who was Desiderata supposed to marry? and (2) Did the marriage ever take place. For the first question, on the one hand, we know that Charlemagne was (or would at least later be) interested in having positive relations with the pope. On the other, the pope actually lavished further letters on Carloman, consoling his wife and offering to baptize her son. I think these facts suggest that Charlemagne was not actually the focus of the pope's concern, and that Carloman was actually the main person of interest in marriage negotiations with the Lombard family.

For the second question, the earliest report that a marriage actually occurred between Desiderata and a Frankish king is found in Einhard's Life of Charlemagne, written sometime after Charlemagne's death, perhaps ca. 817. In a summary chapter of Charlemagne's marriages, he includes "the daughter of Desiderius" in his list. There's good reason not to place too much trust in his list. Einhard's list of Charlemagne's family, although it's the most complete, varies slightly from other sources. He seems to have had the best access to the royal archives, which he used to piece together Charlemagne's early years before Einhard came to the court and could count himself as an eyewitness to events. Nevertheless, Einhard wasn't terribly interested in Charlemagne's early years, and he gets some facts wrong in the early chapters. His supposed marriage to Desiderata could be another such case. It's easy to imagine him seeing the letters from Pope Stephen, cross-referencing the Royal Frankish Annals, and then making the assumption that these marriage negotiations must have centered on his own hero, Charlemagne, and since Charlemagne never failed, this marriage must have come to fruition.

But on the assumption that Einhard's account is accurate, that Desiderata did make it all the way into Francia, that she was joined to one of the brothers in matrimony, and that the chosen brother was in fact Charlemagne, what about divorce? (Pinging /u/TerryStedman) Quite frankly, it seems like divorce was fairly common among the Frankish kings, as was keeping mistresses and concubines. Charlemagne had several illegitimate children through these kinds of relationships, and once his son Louis the Pious took over the empire, he made sure that all his illegitimate half-siblings were handed over to the church, where a few of them became important abbots and nuns.

Looking back at Pope Stephen's letter, he seems to have discouraged this kind of behavior on principle (it's not permitted to dismiss your wives) but he actually seems more concerned with the reasoning behind it (so that you can take other wives who will mix you up with a foreign people). To me, this sounds like a tacit acknowledgement that the church could not forbid divorce; that if the church wanted to have an influence over Frankish politics, it needed to accept that Frankish marriages were part of the political structures and were therefore much more fluid than a good bishop might like to see; and that every time a wife or consort was set aside, her children could be effectively conscripted into the church, strengthening ties between the royal family and ecclesiastical power.