If you’re doing a Lady Gaga costume for your baby, the possibilities are really endless. In fact, at this point, because Lady Gaga has had so many different looks, you could do anything weird with your baby and claim they are actually just dressed as Lady Gaga. (I.E. the Venom costume could work as the Lady Gaga costume, too, just circa her ArtPop weird conceptual phase, right?)
The point is, choose your favorite era of Gaga, and just go with it. We would recommend something silver though? Maybe a blonde wig? Babies look great in blonde wigs.