If your 68 is anything like my 66 the short answer is that it will probably fit but it's a bad idea.
The convertible frames are boxed in - like a full square/rectangle - where-as non-Verts are shaped like a 'C'. 3 sides of the square/rectangle shape. The Verts really need that for extra strength since it doesn't have the structural hardtop. You put a regular frame under a Vert and it will have more twist and lousy torsional rigidity. Things could go bad quickly.
Also, I believe our Verts have an extra set of body mounts in the back trunk area that non-Verts don't have. You'd need to fabricate that into existance on a non-Vert frame.
Here is the frame mount diagram for a 68 for current and future reference:
You’d have to add a couple of mounts, and then close the frame with another strong brace to make it all work. Not a big deal for a full fabrication shop, but enough to scare my shade-tree behind from taking on.