Neomycin resistance gene in E. coli stb3 cells
PGK is mammalian (murine) phosphoglycerate kinase promoter. It will therefore not work in bacterial systems. Why don't you select using ampicillin only? You can increase the ampicillin concentration if you want. 

Regarding:


  I can't find anyone selecting bacteria by Neomycin anywhere.


Kanamycin is generally used for bacterial selection. The Neomycin resistance gene works on Neomycin, Kanamycin and G418 (a gentamycin variant). The gene basically codes for an aminoglycoside phosphotransferase and different variants of the gene have different affinities for different aminoglycoside substrates. For a basic reference see this.