The other approach - buy an early to mid '80s sport bike (Japanese or the like; once the Japanese paved the way, virtually all the major players got on board with similar tubing. threads, fits, etc). Horizontal dropouts. All you need for a really good single speed/fix gear road bike. Good brake options, front and rear in case you ever want super climbing handles on your handlebars. (Might save your butt some day too.) If you bbuy one sized for 27" wheels, you will bet fender eyes almost certainly and have plenty of room for big 700c tires and fenders. Might need longer reach brakes. Mafac Racers are still out there, work great and expensive modern copies made. (They work great long, long past looking respectable so buying unseen is not a very risky. Quite a lot of pivot play doesn't matter and Renee Herst and others make new bushings.)

This from a guy who's ridden fixed: a Peugeot UO-8, a wonderful sport Japanese Schwinn, a mediocre Sekine, a Miyata 610 that went forever through a lot of abuse and my current ~'83 Trek 4something. And my custom Mooney with its horizontal dropouts is now fixed and a sweet ride. I ordered my custom ti fix gear with unique more horizontal than usual "road dropouts": very long and opens at the front down, not forward. The adjustability of a very long track end and joy of horizontal/vertical for wheel changes and flips.