Lol, I meant it when I suggested check out what happens in history with regents for nobility . . . the de jure power of the regent becomes, at some point, official de facto power of the regent. King Henry VIII selected his in-laws to be Edward's heir for a reason- they could not gain the power of the Kingship in their own names, but by ensuring his survival and being crowned it would be the means to their having access to power & it's economic benefits. For an example of how it goes badly you merely need to look at the infamous case of the Two Princes of the Tower-
https://en.wikipedia.org/wiki/Princes_in_the_Tower. The longer such a regency exists, the more likely the regent is to take the actual power into their hands. BT's own regencies have problems- look at what happened with the Camerons, it is how Aramis was able to reach the levers of power due to a vacuum around the throne. Kerensky failed in his duty as regent, he should have handed over the regular operations of the SLDF when he became Regent- it was no longer his primary duty.
The Karosas regent could even be more zealously following Kamea's instructions in the hope of ascending the vacant seat. Besides . . . overseeing the official investigation into finding possible heirs is a great way to to either disqualify candidates or make sure they have tragic accidents . . .
Trinity Alliance did not help the New Colony Region- they LOST Detroit with it's industry and just restored mech production. The area was a economic doormat to the Trinity powers w/o any recourse or political leverage. Sun-Tzu was interested in restoring everything to the Capellan Confederation they once owned, some of the planets in that region were under Cappies auspices during the Star League era. They would not have received the attention of the Chaos March, but the Trinity Alliance allows all parties to extend spheres of influence into the un-aligned worlds without the risk of setting off a neighbor.