Consider a metal rod. You put your ear to one end. Someone "pings" the other. You will hear the ping thru the metal. Now add a wad of cotton balls at your end and repeat. Press your ear firmly on the cotton. The sound will be muffled if it comes thru at all.
When your vibrating exhaust is attached directly to the car, the car picks up and can amplify the sound. To avoid the problem, you must decouple the vibrations generated by the exhaust from reaching the car. You have two approaches. You can add the equivalent of a non-transmitting medium between the exhaust and the car body. Rubber acts as a vibration isolation medium. An appropriately designed spring can do the same. The other approach is to reduce the contact area between the exhaust and the car. By example, at the point where the exhaust runs under the car, hang it by a strong-enough wire rather than attaching it with clamps to the car body.
This is the theory. The best practice is likely found by researching trade journals or chatting with folks from car shops that do the installations.