Convertibles also require the doors be in place to keep the body from flexing itself apart. They also have extra rocker reinforcements and X-bracing under the unibody rails (if you would like, I can take a picture of the underside of one, so you can see the difference).
If you do not care about the car, and it is going to be an "off road only" total loss, then it won't really matter much. You can bend and twist the body all you want. Put ripples in the roof, crack the dash, break the A-pillar brace, crack the glass, all that fun stuff. But if you want to be able to actually continue to use the car ON the road, then it is not a good idea.
To illustrate how important the doors are, however, go ahead and jack up one corner of the car by driving one front wheel up on to a ramp, and park it there. Then go and open and close the doors. You'll see how much the body flexes (in some cases, you may not even be able to open the doors, or if you can, it takes a LOT of effort, and you will not be able to close them again).
Your car, you can certainly do what you want to do. You asked, I answered.
As I suggested in my previous post, you may be able to offset the lack of structural rigidity by the loss of the doors by replacing them with some sort of cross beam...like a sideways Y or maybe K or X shaped brace that goes between the door latch clasp on the B-pillar base and forward to the hinge mounting points at the A-pillar base. It won't be as good as the whole door, but it will be better than nothing, and you could perhaps make it so it could be climbed over for ingress/egress.
