There are plenty of classic/vintage mixte bikes out there, most are mild steel, even a few high end made with Reynolds 531 or Columbus steel. They are typically slow to sell & therefore low cost.

Buy one, have the dropouts spacing cold set to fit modern wheels, then build up with new components, what ever you like. Final cost can be lower than buying a new frame, but you are only saving frame cost. A truly low cost build will require finding lightly used, bargain components. Don