In the Kapteyn discussion, another thing to remember is that Janos had suffered through a civil war, launched by his closest brother and roping in one of his sons, that was initiated almost directly by Maximillian Liao. Precentor ROM was there to push the pieces around the board, but it was Mad Max who put the bug in Anton's ear that Janos was doing it all wrong and it was Max who put the entirety of Wolf's Dragoons into Anton's order of battle. Janos might have been bothered to do more than the four planetary strikes of Operation DAGGER if Max wasn't poisoning the pot of the alliance. The fact that the Combine needed help didn't matter when Janos personal mortal enemy was also asking for help. It made more sense to do the bare minimum all around and let his enemies destroy each other.
As for the League/Lyran relationship, aside from the quote about shooting Steiners that we've all been bandying about, there's another good example of the simmering boil of this relationship: Janos' reply to Katrina Steiner's peace offer. It's been a while, but I'm pretty sure it amounted to "You marry one of my sons, the Commonwealth becomes a vassal state of the FWL." He wasn't proposing a merger like Hanse did, he straight up demanded the dissolution of the Lyran state and its incorporation into the League. That's pretty dang harsh.
Additional note: I've long had an idea to write an in-character military treatise on the history and operations of the FWLM though I never really did get any steam under it. But in my initial round of research, one of the biggest factors that jumped out at me was that in every era, every Succession War, the League gained worlds at the expense of the Capellans but had a net loss of worlds at the hands of the Lyrans. Declining to attack a foe you know you can beat is one thing, but allying with other powers against the one neighbor that constantly gives you a beating is actually smart. You know, if you actually follow through on the terms of the alliance.