Personally I think the Chesterton worlds are going to be something of a problem for the CC (And I've said this before). The CC has long claimed that the people living on them are Capellan Citizens, which has a very different meaning to other nations, so either the people living there are going to be made Citizens without having to perform the customary service, which is going to upset people elsewhere, or their not going to get it, which is likely to upset people living there more then is normal for being conquered.
As Iracundus rightly points out, the Capellan view is that the worlds themselves, in the geographic sense, belong to the Capellan people. The people who currently live on the worlds - and whose ancestors have lived on the worlds as loyal members of the Federated Suns for longer than the Capellan Confederation has even existed - are squatters. As such it is
ideologically unproblematic if those people reject Capellan rule. The Capellan political philosophy here doesn't care about the will of the people, so they don't need any popular support to claim legitimacy. After all, they've been claiming legitimacy for eight hundred years or so without one whit of meaningful popular support.
The problem for them is purely military. How do they pacify Chesterton? The options, as you say, are either give out citizenship
en masse and hope that this achieves something (I think it is unlikely to, but they could try it), at the risk of discontent elsewhere in the Confederation, or to establish a military government by force. That seems likely to stir up immense discontent, but the Confederation might theoretically be able to do that.
Realistically, I don't think either course is particularly important in the short term. Both potential courses of action assume that the Confederation will be able to hold Chesterton for the immediate future, and based on all previous history, I don't see that happening. Chesterton is too deep into the Federated Suns and too core a world to be lost over the long term. We've seen similarly deep pushes in the Succession Wars before and I'm not convinced this is tenable. The current 3145 setting is a cliffhanger: none of the current battle lines are stable borders.
With both the Federated Suns and the Republic gearing up for large-scale counter-attacks, if I was a Confederation planner at the moment, I would be more worried about holding Chesterton than I would be about integrating it. To be honest, I would advising the Chancellor that we almost certainly can't hold it, and that it would be better to use this opportunity to focus on consolidating more defensible gains. That said, I wouldn't be surprised if there is immense political pressure to hold Chesterton for its symbolic value...
Though if I were a typically sneaky Chancellor, I might privately agree with the general, especially since large-scale resistance on Chesterton would make for excellent anti-Capellan propaganda. If the Suns take Chesterton back, that relieves us of the problem of integrating it, and because of Chesterton's symbolic value, it seems quite plausible that the Suns will tunnel-vision on it. So I might expect to lose Chesterton, but I'd try to set it up so that the Suns take as long as possible doing it, giving me the maximum amount of time to integrate other territories. Then once we do lose it, we can put out all the standard propaganda about the vile Davionistas. If I'm the Chancellor, I like having enemies that look powerful. Nothing rallies the Confederation better than fear of a common enemy.
So where did the Kansu Braves, Bharatiya Ganpats, Chesterton Long Rifles, and Hamal Cossacks come from if not from Kansu, Bharatiya, CHesterton and Hamal? Giving money and weapons to them doesn't make them any less from these planets.
There are billions of people on Chesterton, and the Confederation has a very strong interest in supporting the line that the oppressed people of Chesterton long to rejoin their brothers and sisters in the Confederation and so on. It seems entirely appropriate for the Confederation to name a few units after those planets, particularly since it reinforces the Confederation's own sense of having been historically wronged. Continuing to prominently display the name 'Chesterton', and staff those units with whatever number of Chestertonian malcontents they can recruit, is an excellent way to support the Confederation's sense of historical victimhood.
If that makes sense? The Confederation's people are particularly devoted to the military and supportive of Capellan military goals because of the idea that the Confederation has been wronged. It was invaded, unprovoked. It was morally pure and innocent until greedy, thuggish neighbouring states tried to steal its land and enslave its people. The more reminders you have of that narrative, the better. 'Ancestral Capellan worlds' is one of the best justifications for war they have, and the people must never forget how much their neighbours have stolen from them.