My initial response to the script leads me to believe that a suggestion of the period would be very important in order to support both the realistic action of the radio show studio as well as the world of the imagined show itself. It seems to me that there needs to be some kind of distinction between these worlds, one that may most easily be made by lighting, but also by careful selection of costume elements for the story's characters. The characters in the radio show within the play are stock characters, and should be easily recognizable as such. Perhaps this radio show is not completely realistic - maybe they are somewhat heightened or stylized in the tradition of Film Noir. One could even go so far as to restrict the color palette of these characters to give them a stylized film flavor. And in contrast, the real world of the radio show studio might have more color, or a more complicated, fleshed out line.
Costume transitions are challenging, and would need to be simple in order to be accomplished as outlined in the script. Careful attention to detail in the selection of these costume items would be necessary in order to chose pieces that would exemplify the characters themselves and communicate this to the audience. 1940's costumes have a very recognizable silhouette that can easily be achieved through the use of pulled or purchased pieces, and supported visually with distinctive hairstyles and makeup of the period. Some costumes would of course need to be built due to the nature of what they need to do onstage - such as Charlotte's costume that needs to break down onstage during her dance number.
The Songbirds almost seem to function as both chorus and background, helping to disguise transitions and establish locale. Without speaking to the director and knowing how he views them I would say my initial response it to use them like the chorus of dancer/singers in Sweet Smell of Success, outfitting them in base costumes that reflect the dance clubs they tend to inhabit such as tuxedos and cocktail dresses, and keeping them within the feel of the imagined radio show world. Hats or coats could be added as needed to help establish different locations.
Vintage Clothing Sources -
Film Noir Films (warning, popups)