Do you have friends in the States that can bring some over for you?
You wouldn't need much. I think I may even have a bit here somewhere.
I still have some invading small patches around here but most has died out. I originally dug very very carefully in the area it was growing the most in and ever so gently worked at getting the largest part of the root out of the ground. I also took out as many little pieces of broken off root I could find. Time consuming yes but worth it if you have some plants that mean a lot to you.
If you think you are going to lose that bed could you move the roses you want to keep? Prune them way back and try to move them to a different area? Then you could work on getting rid of the false morning glory. Hot water and vinegar may do the trick if you are willing to move the plants you want to save and then don't care what happens to the bed.