Seems easy enough, yet tricky. The ones that pop to mind are BarbeDisplaySSK, Big City, Black Tie Display SSi, Broad Avenue, Broadway (as you mentioned) and Ritz FLF.
As usual, no guarantee that the list is complete.
For 1950's American Diner Broadway should do but Big City or Broad Avenue are probably the one used (just a bit thicker. The nicest one, I think, is the original Ritz FLF. The text PALOOKAVIIE is a different story. Looks like the others but is not. K and E flipped would help but the P doesn't come close at all.
I'll see what I can find ...
Have fun and ... be nice to the cats
_____________________________________
Relax ... it is just a font