Your Masters degree will be enough for you to legally find work here. A TEFL cert really needs to be at least 100 hours to be of any use - and have a teaching component, which the short courses generally don’t have. They are usually theory only.
Having said that, when I first started teaching, I initially did a short 40 hour course. The reason I did the short course was because I was working in corporate IT at the time, contemplating a career change, and didn’t really have enough spare time to commit to a 100 hour TEFL. It served to give me a vague idea of what teaching was like, and was a pretty good review of all the basic grammar I’d learned at school. It wasn’t much else. And didn’t make any difference to finding work in Taiwan (although it was all I legally needed for working in Vietnam).
If you have some experience in teaching (music in your case), you should have some foundations for things like time management, lesson structure etc. In my case, I had a little experience in teaching English, gained during a short volunteer stint in Sumatra, plus I had a fair amount of experience in workplace training (mainly training adults on database systems), so those helped me more in Taiwan than the short TEFL course did.
Please excuse any spelling/grammar mistakes - I haven’t finished my first coffee of the day and am still half asleep.