Success!
Just completed hauling the bugeye to the fabricator and everything worked perfectly.
Thanks to ChrisS for bring the 2 different styles of Uhaul auto trailers to my attention. I ended up getting the painted (older style) hauler, and the inside to inside measurement of the lips of tracks were 36". The rear (smaller end) of the BE is 39", so it sat a little bit on the lips of the tracks, but just a touch, and once I ratcheted down the front tire webbings, it was very secure.
BTW, tire webbings are a good set up, as you are only using the tires to secure it down--not wierd hooks and contraptions that are supposed to hook around the front or rear end--which I just spent countless hours painting, polishing, etc, etc. Also, part of the reason the car is going to a fabricator is to correct my steering rack cross bar which was initially bent by the flat bed tow truck guy who used it as a bar to secure the car down on top of the truck...more reason I like the tire webbing!
Again, this was with the old style, orange painted Uhaul trailer. I do believe the new galvanized trailer (non-orange) will not be appropriate, as it is about 4" wider.
Thanks again for all of the input, and hope this thread helps folks with this predicament in the future.
Mustafa