Alright, let's go with that scenario for the sake of argument. What would be the casus belli?
If it's something like rare elements available only in the Oort cloud, the traditional colonial model would apply and the overwhelming interest in the Oort cloud would be in those very rare elements and not as some sort of refuge. Your Oort cloud pilgrims would have to be fiercely protective of their rare elements and to have gotten there first before the rare-element prospectors. The Oort cloud would have to be not big enough to satisfy both groups of people, and the Oort pilgrims would have to be stubborn enough not to move their replicator-colony to the Kuiper belt or elsewhere.
At that point, you can start to reasonably speculate: you have clear objectives on both sides at the very least. It becomes very hard to predict what would happen because we've stipulated so many counterfactuals that everything we know would be obsolete. Logistics, which has dictated almost every war in human history, would look totally different due to the fabrication technology, though limits to propulsion technology and the vast distances involved would cause other issues. The Oort pilgrims would have the advantage of locality--they could see and respond to any movements from the inner solar system within months while delaying or hiding their defensive actions until the last minute. But the fabrication technology would render pretty much anything expendable, and it might be feasible to turn maybe a few hundred Kuiper belt objects into a swarm of self-guiding missiles or something while the Oort pilgrims wouldn't have enough raw materials to really hurt, say, Earth.
If it's something like rare elements available only in the Oort cloud, the traditional colonial model would apply and the overwhelming interest in the Oort cloud would be in those very rare elements and not as some sort of refuge. Your Oort cloud pilgrims would have to be fiercely protective of their rare elements and to have gotten there first before the rare-element prospectors. The Oort cloud would have to be not big enough to satisfy both groups of people, and the Oort pilgrims would have to be stubborn enough not to move their replicator-colony to the Kuiper belt or elsewhere.
At that point, you can start to reasonably speculate: you have clear objectives on both sides at the very least. It becomes very hard to predict what would happen because we've stipulated so many counterfactuals that everything we know would be obsolete. Logistics, which has dictated almost every war in human history, would look totally different due to the fabrication technology, though limits to propulsion technology and the vast distances involved would cause other issues. The Oort pilgrims would have the advantage of locality--they could see and respond to any movements from the inner solar system within months while delaying or hiding their defensive actions until the last minute. But the fabrication technology would render pretty much anything expendable, and it might be feasible to turn maybe a few hundred Kuiper belt objects into a swarm of self-guiding missiles or something while the Oort pilgrims wouldn't have enough raw materials to really hurt, say, Earth.