I don't know much about brome to be honest. I know it makes great hay and it's common in ditch mixes where I grew up. I wouldn't kill it. That could touch off the weed apocalypse next year. What I would try, is coming in with some durable flowers like red clover, white clover, and hairy vetch. I'd try throw and mow on it late in the season, like 4 weeks before frost.
It likely won't be great, but I've seen hairy vetch muscle its way into my quackgrass like a boss, and that stuff flowers all season it seems. The white clover and red clover could/should start showing up in clumps.
My lawn at the cabin was solid untouched quackgrass when I got there. I got to work clearing trees, pulling stumps, and put in the cabin. To make my lawn, I picked up the sticks and just started mowing the quack grass. I never planted anything, or amended anything. I just mowed what God gave me. That was 5 years ago. Today, my lawn is a destination clover plot for the neighborhood deer as it seems it's almost 50% white clover. There wasn't a blade of clover until I got to mowing and mowing.