Sun-Tzu was less a "compromise candidate" between Katherine and Victor and more a ploy by Katherine that backfired horribly. She wanted to be First Lord (because, well, that's Katherine for you), so nominated Sun-Tzu, knowing Victor would object vociferously, she could mediate and come out looking like a sane and reasonable peacemaker. Unfortunately for her (and fortunately for just about everyone else) Victor wasn't quite as stupid as Katherine thought, and said what amounts to "Yeah, I don't like him, but I've got to go off and fight the Clans and the position is 100% ceremonial anyway, so it really doesn't matter to me." Katherine was shocked, Sun-Tzu was elected, and the operations went forward.
And yes, the seat of First Lord was 100% ceremonial by design. If it had any actual power, it would just be more Succession Wars status quo as each House Lord would think only they deserved to wield that power (and even the ceremonial nature of the post was something every House Lord coveted deeply, and led to no small amount of infighting and wrangling). Sun-Tzu had zero influence over the military operations conducted under the banner of the Second Star League, though that didn't stop him from claiming some PR credit for the successes. Anyone with half a brain knew Sun-Tzu had nothing to do with the military success of the Second Star League against the Clans, but to the Capellans (who, yes, do have half a brain) it was still a great symbolic victory, that their Chancellor had been "in power" when the threat of Clan Invasion was ended for good. And even if he had nothing to do with, history books for a thousand years will note that Sun-Tzu Liao was the Elected First Lord of the Second Star League at the time the Clans were decisively defeated. His name will live forever for that alone.
As for the Chaos March and St. Ives. . . the Chaos March was a poo production well before the Second Star League, so quelling the Chaos could still net him political points. I don't know a lot about retaking the St. Ives Compact, but from what I understand, he didn't use Star League troops to do it (he used his own Capellan forces), and the Second Star League did try to intervene, but since it was a psychological warfare tool against the Clans and not a functional governing or mediation body, it had functionally zero ability to do anything about it, declaring it an internal Capellan matter because there was nothing else it could do.