It's certainly possible. You will have to cut away about half of the galley and take a large chunk out of the door-side closet. You need to maintain adequate support for the upper shell. Doing it the quick-and-dirty way will impact the resale value of the trailer down the line.
For less work, consider making the bed 60" at the head end but less at the feet. Also, consider whether you really need a full 60". Even a 54" double is a significant improvement over the standard 45".
Another approach is to make a bed extension in the center aisle and sleep longways with your feet in the aisle.
In my mind, a 13'er with a really large bed at the back would ideally have a front galley, which means it's pretty much a gut-and-rebuild involving rewiring and replumbing along with mostly new cabinetry. The time and cost will be significant, but if well done could enhance the value of your trailer. Member Ian G's Boler
comes to mind: Project Boler
As an alternative, consider selling the 13' and looking for a Casita
Liberty 16'. It has a huge, almost-king-sized bed oriented longways with a full galley and front bath. It's not the most common model, so it may take some patience. In the big picture, considering your time, it might be cheaper than making major modifications. Your 13'er should be quite saleable, especially if you wait until the spring.
Figure a 16' trailer will weigh 2500-3000 pounds, so your tow vehicle comes into the picture.
One last thought... My wife and I find the standard bed comfortable for couple time in the evening, watching a movie together... But when it's time to roll over and sleep, I move to the front bunk. Then I don't have to disturb her to use the bathroom during the night, or when I rise in the morning. She likes to sleep later, while I like to go outside, make a cup of coffee, and enjoy the early hours.
Sometimes the best way is to modify yourselves rather than the trailer.