For what you described, it only works if you have two kinds of chickens that are identical except for whether they have white barring (necessary for autosexing) or not.
With Bielefelders, I don't think there is any other breed that is the same except for the barring.
If you want to introduce new genetics while keeping the barring gene, you can cross a barred rooster (like Bielefelder, Barred Rock, or Cream Legbar) to any hen. All daughters will be barred, and can be used for breeding just the same as if they were purebred from any barred breed. (This only applies to the barring gene, not to any other genes those hens may have.)
Genetic reason: barring is on the Z sex chromosome. A rooster has ZZ, one inherited from each parent. A hen has ZW, with the Z coming from her father and the W from her mother. So a hen can only have one copy of barring, because she only has one Z chromosome. She gets it from her father, and passes it on to her sons, but it cannot travel from mother to daughter.