Any answer that goes along with this question is based upon conjecture. We lack the necessary information to give more accurate answers and to give a more in depth answer I would definitely need to take more time to research the topic, which I now intend on doing.

I am not going to be able to answer the question directly, but will instead try to cover what the result of an alliance between the two leaders would have meant and if it would have been beneficial. There are a few background points that we need to understand prior to coming to our conculsuon. The most important of these points is the position that Constantine III was in during Alaric's war in Italy.

Constantine III was a general who had been declared Emperor by his troops in the Imperial province of Britain. He then moved the remainder of the comitentensis troops from the Island to Gaul to repel barbarians from the Rhine frontier. A number of leaders in Hispania also swore fealty to Constantine. From this we see that he controlled on paper a large territory. Britain, Gaul and Hispania made up roughly half of the territory within the WRE. Only Italy, Raetia, Africa and the western portion of the Danubian frontier we're outside of his control. This assumes that he had a large number of taxable citizens, the vast recruitment potential from the Rhine frontier and strong food supplies from the vast territory controlled.

The reality was that Constantine III faced a number of weaknesses that cannot be understated. One of the largest issues was the Rhine frontier. What had centuries ago been an impressive series of forts and defenses had by this point in time fallen into a terrible state of disrepair. Tribes had been ignoring the defenses and raiding deep into Imperial territory to great reward with few consequences. The cities of Gaul were under constant threat of invasion and many of the villages and towns were plundered. The potential for a good agricultural yield reaching the camps of Constantine would have very likely been significantly reduced. Furthermore the tax policies of the late empire would have vastly declined any potential to bring in good revenue without being supported by trade to the East. Given the position of his territories it would have been difficult for him to gain trade and even more so because of the presence of hostile forces in both Italy and Africa. Lastly he lacked any sort of legitimacy. While the soldiers believed him to be a decent pick in their moment of crisis, there was nothing preventing another from claiming his title once the circumstances were changed. As we know this is eventually how he met his demise. Constantine would have been very aware of the fact that he lacked any claim to the Imperial office, whereas Honorius was a member of the Theodosian dynasty. All of these factors leave Constantine in a precarious position. He could never be sure that barbarians were not about to invade and attack his rear. While he had a sizable army and decent generals, he could not be sure that the situation would hold.

Alaric was also in a difficult situation. He had long depended on Imperial recognition to maintain his people's place within the Empire. There are definitely benefits that making an alliance with someone like Constantine could have provided. Constantine could have provided the Imperial recognition if sole Emperor. He would also be able to give the Gothic people a high standing within the Empire and good lands to consider home. I find it likely that if presented with the opportunity that Alaric would have been happy to at least consider an alliance with the rebel general.

In 409 Constantine was prepared to invade Italy and Alaric was currently invading Italy. If the two were to have created an alliance, this would have been the time. However you have to consider what Constantine had to gain and what he had to lose. If he sided with Alaric to overthrow Honorius and become the sole Augustus, and then eventually appoint his son most likely as co-emperor, he would become master of the Western Roman Empire. He would likely provide great lands and titles to Alaric and his people. And he would still face all of the threats that faced him. Constantine would not be able to hold the Empire on his own. If he stayed in Italy to eliminate Honorius' allies, it would likely result in losing the rest of the Empire. If he was to defeat Honorius and return to Gaul, he would not be able to hold Italy. And regardless of either situation, the generals of Africa would likely refuse to send grain to the usurper. But most importantly he would still lack anything to cement the legitimacy that his family desperately needed.

On the flip side, if he stopped just short of destroying Honorius, he would be able to take the position of co-emperor. He could return to Gaul knowing that Italy would remain open to him, the grain of Africa could then supplement his dangerously low supplies and knowing that he could focus on forcing out the barbarians that had invaded when he was in Italy. It wasn't a perfect solution, but Constantine didn't need a perfect one, just one that would keep his family in power.

Alaric had troops that Constantine could have used, but Constantine already had plenty of troops to defeat Honorius. The sitting Emperor's armies had abandoned him and all of Constantine's were still loyal. He almost had no benefit in allying himself with the Goths.

Furthermore, there is another factor to consider. Constantine's armies were like any Roman force during this time. He would have had a large contingent of Foederati. Yet these would have consisted of people's such as the Franks or the Alemmani rather than Goths. If Constantine brought on the Goths as major allies it would likely give great concern to his current troops and generals. It could have possibly opened up the threat of a usurper being declared.

It is for these reasons that I do not believe that Constantine III would have formed an alliance with Alaric to topple the existing Emperor. There was too little to gain and so many potential threats, whereas becoming co-emperor was simply a safer choice. As we know, this did not work out for Constantine. One of his major generals did rebel and eventually killed Constantine. Yet, we cannot use what happened to assume the decision-making process that Constantine went through. There is likely very few alternate choices that would have resulted in his survival. Geronius was one of the major generals, his troops were loyal to him personally and be was obviously connected enough to gather support in Iberia. Given Constantine's own rise to power, it is hard to believe that Geronius would not have considered how easy it would have been to have his own loyal troops declare for him. Like Constantine had been, he was a strong leader in an area that was isolated from the center of power.

We can never know what would have happened if Constantine had allied himself with Alaric, but I do feel that this analysis been enough to gain a very basic background into the considerations that Constantine would have had to make.

Please let me know if this has been helpful. I am absolutely love late Roman history and am more than happy to debate any concerns that you might have about my answer.

Thank you.