Can I transplant a mature hibiscus? If so, how and when?
You can and early fall is a great time to do this. Don't wait too long into fall or else the plant will not have time to establish before cold weather sets in. So if you cannot do it this fall, spring is also a good time.
Also, it may help to do it in the evening to reduce stress/transplant shock, and be sure to include as much of the roots as possible. This article will help avoid transplant shock: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm
I would also trim the plant back by 1/3 right before you move it, as this will improve the chances of the plant's ability to survive the move.