Bret knows Liam is lying about cheating on him but can't figure out why and it's driving the man up the wall. It obviously has something to do with the work trip that so obviously fucked him up but he doesn't want to pry and make shit worse. And he meant what he said when he essentially told LIam he's a control freak. He loves the man so much but enough is enough. Bret has his limits, and Liam went over the edge.
Bret's going to stay in Austin because another move across the country sounds exhausting, especially when this is potentially fixable, and he doesn't want to permanently leave Liam. God help him, he still wants to marry the man. It's cheesy to think it, let alone say it out loud, but William Walker is the love of his fucking life. The thought of throwing away everything they've worked so hard for makes him sick.
Liam will crack at some point - it's inevitable - but he'll have to make the first move (and beg for forgiveness) if he wants to fix this. Bret doesn't want to move on but he's also not waiting around either so he's stuck in limbo.