score:2
This is the use case that the two-entry visa is supposed to solve, so if you present your entire itinerary (including Romania and the return via Budapest) in your application to the French consulate, they should be willing to issue a two-entry visa that allows what you want to do.
Note, though, that if your return journey is by plane from, say, Bucharest with just a short layover in Budapest with your baggage checked through, you may not need to enter the Schengen area on the way back at all, and then a single-entry visa will suffice as far as Schengen is concerned. This will not be to your advantage, because then you would need a separate visa for Romania whereas holders of two-entry Schengen visas can enter Romania without a specificially Romanian visa. (And the entry into Romania doesn't count as one of the "two entries").
So it would be in your interest to structure your plans such that you do some real visiting inside the Schengen area after Romania.