This has come up before. While it
is possible, it likely wouldn't make for a happy seahorse. Your refugium isn't tall enough to keep a seahorse truly happy -- most species need a swimming height of 21+ inches; the taller, the better. Besides, the point of a refugium is to provide a predator-free environment, and introducing a seahorse in there would only reduce the copepod & amphipod population. Finally, unless your fuge is visible as a display, it seems a waste to put such a beautiful animal in a hidden environment like that.
Your best bet is to start looking for a new aquarium. Seahorse tanks are not difficult to set up, and you can supplement the horses with other slow moving or timid fish. You can also add quite a few coral species into the seahorse tank, making it essentially into a nano reef. Just make sure you do your research, and don't forget to get a tall tank!