In short, it would have been messy.
But this is my favourite WI, so I'll give you the long version:
Karloman's victory in such a war will depend more on Charles' other enemies bailing him out than him being able to beat his brother. The good thing for Karloman? Charles has a lot of enemies. The bad news? They are all pretty far away and he won't be able to speak with them all that easily. Let me break it down:
Aquitaine: Charles Martel, Pippin III and Charlemagne all had this habit of making the Aquitainians not like them, and there was frequent rebellion in the area, the last of these being spring 769. The big division between the Karlings really began here too, as Karloman refused to help his brother put them down. Aquitaine was going to support Karloman in a war simply because he had more or less let them be, while Charles had been the one capturing cities and demanding the heads of leaders. I don't think Hunald was alive any more, and Waifar certainly wasn't, so the biggest leader in the area is Lupo of Gascony. His lands are small though, so I'm not sure he would risk it unless he thought he could win. Even if he did, it would have little impact on the war, and be crushed shortly afterwards if Charles won.
Saxons: Karloman or no, Deventer (some Dutch town with a church) will still get sacked by the Saxons in January of 772, which may even be before Karloman and Charles start fighting each other. Charles was quite pious too, and would almost certainly react (he didn't order the Blood Court of Saxony without some good reason after all). Saxony is quite powerful and certainly hostile to Charles, so that represents a good ally for Karloman. How will Karloman coordinate his movements with the Saxons though? They are quite far away and don't even really have a centralised government, so I'm not sure they will do enough on their own. Certainly better than nothing though.
Bavaria: Duke or King or whatever the hell he was Tasillo III is a notorious oath breaker. He was technically Charles' vassal in the division of 768, but I can't see him just going through with that because 'it's noble' or something. Bavaria however is a powerful, centralised kingdom that can be a real thorn in the side of whoever he declares against. I don't imagine him siding with Karloman just because he wanted to stir the pot, so he could go either way, but more likely he remains neutral. The most important part of this is that Bavaria drifts out of the Frankish orbit for a good 20-30 years at a minimum.
Lombards: Like Bavaria, they are strong and unified, and unlike Bavaria they have a good reason to support Karloman. Any army they raise and send over to help Karloman will be an important part of him securing a victory.
There's just one problem though: The Lombards do not like the Pope. They have been trying to invade the Pope's lands for a good generation or so. What is stopping Desiderius from using the Frankish crisis to grab Rome for himself? Lombardy might be an ally, but it is an ally that has its own agenda.
Merovingians: You look at me and say WHAT?! In 772, there was quite likely still a Merovingian alive. Childeric III's son Theuderic was born sometime around 750 and spent his life in a monastery in what is now Normandy. No further record exists of him, but I imagine that he probably lived into his twenties at least. If this fellow has any idea of who his father was, he will know that he has a claim to the throne that is just as good as these no-good Karling brothers. Suppose he busts out of the monastery. That could mean a lot of trouble if he gains followers (and with Charles and Karloman having lost the Frankish equivalent of the mandate of heaven, he will likely attract a fair few), and with the monastery being part of Charles' kingdom, he will go for Charles first.
At this point though, I say that the Karlings will likely band together to wipe Theuderic out, if he does crop up. Which could go anywhere.
Another point to consider is the fact that Charlemagne's wife on December 4, 771 was Hildegard. She is the granddaughter of a duke named Hnabi, who is in charge of most of Alemannia (about 1/3 of Karloman's kingdom). Hnabi has very good reason now to defy his master and switch sides, which could screw Karloman up quite severely.
So really there are way too many variables to be sure of anything. I haven't even factored in the Spanish Muslims yet, but already we are way too unsure of anything for a conclusive answer. If I had to choose a side, I would bet on Charlemagne winning. The odds would be pretty even though.
- BNC