I should think that public transportation from ORY to Bercy is rather convoluted, time consuming, and requires a series of changes.

You could take the bus to Denfert Rochereau, 8€, but then you still have to reach Bercy. Métro line 6 is the obvious chose but that only takes you to Bercy station, cost 1.90€ plus the time to purchase tickets. I would allow 60 to 75 minutes.

You could also take Tramway 7 from Orly Sud to Villejuif, 1.90€, then Métro 7 to Place d´Italie, connecting Métro 6 to Bercy, for a second ticket at 1.90€ I would allow from 75 to 90 minutes for this route.

Alternatively, you could take the Orlyval, the rail line to which you refer, to Antony, connect to the RER B until Denfert Rochereau and then connect to M6 to Bercy - cost 12.05€ and about 45 minutes.

You could take the shuttle bus to Pont de Rungis, connect to the RER C, at Gare d´Austerlitz transition to Métro 14 to Bercy, cost 6.25€ and about 45 minutes en route.

However, a taxi will cost you a flat 35€ and will be the easiest, if not the fastest option. Do not accept offers from clandestine drivers soliciting from within the terminals. Proceed to the official taxi queue. Travel time is about 15 to 30 minutes, door to door.