It seems with bicycles that almost nothing is ever new under the sun, and trying to find out who did what first is almost impossible. While I was just doing some personal (curiosity) research on the origins of the Mixte frame, I did come across this http://www.rijwiel.net/kruisfre.htm web site that links the Mixte to a group of frames refereed to as Cross frames, or Girder Frames.

If you go to the opening page of this site and look at the hand drawn frame designs, please note that number 33 resembles a Mixte, although it is hard to tell whether it has twin top tubes or not because of the hand drawing. The picture date is 1901, so it does show the history of the Mixte frame going back at least that far.

But, I would be likely to guess that the first "Step through Frame" was more likely a U-frame/loop frame design... But, I will admit I have no evidence to know for sure.