My prediction is that Trillian would take on Brewer and his Vesper Marches first and succeed (with Brewer, annoyingly, again fleeing to live another day), showing that she means business and will try to leverage this to at least make the Hounds and Tamar pay lip service to the Commonwealth.
I think there could be another way out for Brewer, which is hinted in his bio...
"he worries privately about his standing on Hesperus
II, and how he will manage leadership of his new realm and
maintain control over his far-away homeworld."Trillian could offer a return to his ducal position on Hesperus II. Though an uneasy truce, it would work. The Hesperus Guards could be stripped of transports (not a big deal these days...) transformed into a new DSPF, and a tight LCAF security established. Showing mercy and a willingness to recover lost territory would go a long way reintegrating Tamar and Arc Royal.
I mean, they are Kell Hounds... as long as you do not act stupidly against them, they will return to the Commonwealth sooner or later. Same with the
GDL. And then Tamar is a matter of time, too, though some force might be needed.
There are a lot of ways to end well for the Commonwealth. But TPTB seem to have decided the Lyrans are stupid. The sheer number of idiotic Lyran behaviour in this book makes a beneficial solution of the crisis unlikely.
PS: when was the last great Lyran victory? Götterdämmerung in the late 80ies/3020ies? Counting not being annihiliated as 'victory' was irksome for a few decades (irl). Now it is plain boring.