In the past they used to pull private cars. There was a collector who used the LIRR now and then for his Pullman. Once the new equipment came, the locomotives were incompatible with standard equipment. It was easy years ago to simply attach a private car to the back of a scheduled diesel passenger train. Today, they would need to use a 1500 and Roustabout crew. It would have to run "extra", would be in the way and very costly. Or, they can attach the car to a scheduled passenger train, intact with a DE/DM and lead the train with a 1500 (doing the pulling). Why? The DE/DM would be the HEP for the C3's, the Pullman will probably have its own generator and the 1500 does not have a "hold" feature. Negating testing that handle position and allowing it to leave a terminal.
The Long Island could do it if they wanted to. They just don't want to. Besides, our past President, Mrs. Williams stated that 1500's could not be used on diesel passenger trains unless specifically permitted by the GST. I guess they were to "icky". The present GST is "old-school" but why would he bother? He would be up against new engineers and conductors who think that an M-7 and DE30 is railroading and the concept of a handsign is more complicated than the Apollo moon missions, an MofE Dept. who would be screaming about wheel wear and incompatible braking (they are wrong) and managers who would over manage the assignment.
Could it be done? Sure. With the right crew, they could back a Montauk train onto a freight train and run like it's an ordinary day. But who would bother? Do I sound cynical?