To me, a doctor's scale has eye-level weights that can be adjusted for maximum accuracy. Do such scales exist in France? What are they called?
If that's you call a doctor's scale I haven't seen any of those in France for ages. I've had a look around and they can still be bought, they're called Pèse-personne à colonne mécanique. Pèse-personne being the word for "scales", à colonne for the vertical rod and mécanique as "mechanical".
They are called pèse-personne à colonne as indicated by Laure.
The colonne signifies that there's a vertical rod or column, at the end of which the weight measurement can be read. Some "pèse-personne à colonne" are mechanical, some are electronic (based on how the weight is displayed).
The mechanism with the counterweights that slide on a horizontal bar is called "contrepoids coulissants" (sliding counterweights). There are other ways of mechanically indicating a weight, e.g. with a round display.
The full description of what you're looking for is a "pèse-personne mécanique à colonne à contrepoids coulissants".