I'd do as much as possible to get the onward boarding pass at the originating airport, but it might not be possible.
In 2015 my wife and I had a similar itinerary as OP: BUD-CDG-JFK. First leg on AF, second on DL. We did not get BP for the CDG-JFK leg and it was a major hassle to get the BP at CDG.
You'd have to pay me a lot of money to fly through CDG again.