Do aircraft cabins have suspension?
The cabin is rigidly attached to the fuselage, with bolts. But in a way the cabins do have suspension: from the wings riding on air.


Elasticity of the wing construction material acts like a spring. Load it with a vertical gust, and it will bend upwards, then spring back.
The wing bending experiences damping forces from the surrounding air, proportional to bending velocity.
The wing sweep angle spreads out the gust: not all of the wing is accelerated upwards immediately.


Modern subsonic airliners with swept wings, flying with near transsonic speeds, experience much less turbulence than the olden days DC-3 type aircraft. So far, so good.

OPs proposal to provide suspension between fuselage and cabin may not be very effective. Notice that car suspensions work best when the unsprung mass is low: independent suspension is able to follow the un-evenness in the road much more effectively than rigid axle suspension, due to the much lower unsprung mass of only the wheel+tyre.

The unsprung mass of the suspended-cabin would be the whole fuselage-plus-wing structure, engines, fuel etc. Considerably more than that of the cabin plus payload. A lot of extra complexity for not much gain.