National Kaohsiung Normal University asks for the health check as well, but I asked them and they said I can wait until I am in Taiwan this fall to do it. Basically, it would just need to be done before I start classes. They were able to process my application without it for now - so maybe you should just ask NSYSU if they can wait. I’m pretty sure it’s mandatory that anyone studying at an approved Chinese Learning Centre in Taiwan needs the health check.