You've got a lot of trunk on that thing.. Things I ran in to trying to fit my 67 was the trailer I first had was too narrow between the fenders and the dovetail was too steep and caught the bumper. I'd also measure your bottom of door to ground and your trailer floor to fender top. That will tell you if you can open the door to get out of have to dukes of hazzard it out the window. Where are you located?
I have used Uhaul trailers to haul a few 62 Galaxies and a 63 and they fit fine, even with the older style trailers.
Once I drove 12 hours to pick up a car. I went to the Uhaul store and told them I needed a trailer. They asked me what kind of car I was hauling, and I told them a 63 Galaxie. They said that my half ton full size Dodge Ram truck with a class III hitch wasn't enough to haul that car, and they wouldn't let me rent the trailer. I had hauled a few Galaxies with this truck before with no problem at all. So I went to the next closest Uhaul store and told them I was hauling a Toyota. They rented me a trailer and I hauled the car home with no problems.
Once I was hauling a Galaxie with a Uhaul trailer and it was very squirrelly at 50 mph and over. I stopped and checked the air in the tires and they were low. I aired them up to the proper PSI and everything was fine for the rest of the trip.
I have found over the years if you need a trailer once you will need it again. That is why I just buy one and I don't have to worry about getting one when I need it. I only use mine about two or three times a year but it's there when I need it and in top shape.