…..you could always use a chain tensioner or buy a Planet X track frame and drill it for brakes?
You can get drilled forks for the Planet x track, I think they are the same as the tt bike forks. I don’t think you could fit a rear brake, there’s no stay to take one of the keirin rider brake mounts and what’s there doesn’t look drillabe. But if you were going to ride fixed it might be fine if a little harsh and would probably only take a 25mm tire at the front and having just looked probably only a 25mm at the back. also as an all day bike it has no bottle cage mountings.
something like a Pompino but lighter
presumably that’s most fixies then 😉 -what about a Pearson Touche or if you have money to burn get a handbuilt 853 frame or Xacd or someone to knock up a ti bike for you