It will ride slightly rougher without the body bushings, but will be much stiffer. Take a look under some 50s convertibles. They all have "X" bracing in the center. I don't recall how the frame is made on your car, but if it's like the 70s Chevelle frames the rails under the body are very light with heavy rail sections in the front and rear. I'd cut the light sections out and replace with 11 gauge rectangular tubing the same size as the heavier rails. Then make an X section from where the frame starts to widen in front and back. The center will have to have room for the driveshaft. You can leave a 6" or so section out and weld a plate on the top and bottom, leaving the center open. Some of the 50s convertibles did that, with the bottom plate bolted on for easier driveshaft removal.
BEFORE you cut the top off, weld or bolt a brace from the dash along the inside of the car to the rear above the wheel well on both sides, and across the car from B pillar to B pillar. That will keep the body straight while you cut the top off. Otherwise it will warp out of shape.
If you keep the rubber body mounts you will need to add some between the frame and B pillar to keep it stiff. Weld a strengthening bar of some sort to the body between the B pillars in the floor, and extra bracing behind the rear seat for the C pillars. A piece of 20 gauge sheet metal welded to the existing behind rear seat bracing would probably be enough.
It will take a good bit of work to make this idea work right, but it can be done. Will it be worth it though? That's up to you, and what you want from it!