AFAIK, you don't
need to unfreeze your fATPL in order to get a job, it's just a requirement set by some companies. However, that forces you to get
really good test results (>90%) to make up for the lack of experience, and it's still not guaranteed. In my opinion, having a university degree is a nice plan B, in case you don't pass the medical or have any other problem.
I believe it's hard to get a job as a flight instructor when having no flying experience, especially at a renowned FTO, so I'm going to aim directly for a FO job myself. There are several cadet schemes in Southeast Asia; being a Singapore national (or resident, I hope it works the same way) you can apply to Jetstar and Cathay Pacific, and maybe some more that I've missed. Check this out:
Becoming an Airline Pilot | Pilot Training | Latest Pilot Jobs
My advice: prepare hard and try to get in an MPL route (or standard ATPL scheme if the airline doesn't offer the first). If you don't succeed and can afford it, get in a good school and study on your own —some say integrated offers you more chances of getting hired, some say it's not worth the overprice, so it's up to you. If you plan to move to Australia, do some research about local or NZ FTOs.