I have a similar goal as well, I want to teach Psychology (or possibly primary school children) in Taiwan. Where exactly in Asia are you looking to teach?
I am still researching about how to obtain a teaching position at an international school but I can offer what I know. A MAT is definitely one option but there are a few others, depending on which school system you want and grade level. For example, I am very interested in the British International schools and I am looking into getting an MA in Education and then applying for Teach First in the UK to gain some experience. A couple of schools in Taiwan that I looked into specified if you did not complete a Bachelor of Education then having an MA in Education plus teaching experience would make you competitive.
Another option is to get a teaching certification in Social Studies, either by getting a teaching certification in the States or a PGCE in the UK, again depending on what type of international school you want.
Then there is the MAT like you said, which is good for those looking into teaching secondary school and allows for focusing on your subject area and then acquiring pedagogical skills in graduate studies.
I think the fact that you have teaching experience in Asia will help you when applying to schools and teaching positions.
I hope this answered your question, good luck on your path to becoming an international teacher!