Normally, the Customs and Border Patrol (CBP) see the D1D (Crewmember) visa as a blanket crew visa not attached to only one airline operator unless it is clearly specified in the visa stamped/attached in your passport. The specific visas are more oriented to cruise lines and cargo sea ships.
But freelancing in Bizjets is another history, you will need a B1/B2 (Leisure and Business) Visa to enter in private jets to the USA, as you will not be seen by CBP officers as a required crewmember by law, not even if your company operations manual specifies it, this is unless the bizjet is certified for more than 19 seats.
I Suggest to get a B1/B2 visa. Call the US Consulate in your city to make an appointemnt and ask the basic requirements, plus accompany them with a letter from a company where you will work. If you're an EU passport holder this shouldn't be a problem.
On the plus side, having a B1/B2 visa and not being a required crewmember you can always serve as wildcard passenger if using a CBP overflight exemption and none of the passengers on board are listed, or in empty legs this is a great advantage.
All of this being said, every CBP port have their own interpretations of the law and is always a good idea to call your main port of entry to see ho they will see your status entering the US, normally they are a good bunch.
Good luck!