You could apply the 309 offshore and keep working in Singapore or apply for a 820 onshore if your visa allows (no further stay is not good), but you can not work in Australia till your visitor visa stay period expires (normally 3 months) and Bridging Visa A kicks in (No travel rights).
Singapore application (309) might be faster but they seem to require all documents to be certified (only place that does) and that adds to cost and time.
Australian application (820) will take longer, but after application you can get Medicare and after Visitor Visa expires you will have full work rights, you apply for a BVB to travel overseas (in advance).
Relationship evidence maybe you problematic area.
Singapore application (309) might be faster but they seem to require all documents to be certified (only place that does) and that adds to cost and time.
Australian application (820) will take longer, but after application you can get Medicare and after Visitor Visa expires you will have full work rights, you apply for a BVB to travel overseas (in advance).
Relationship evidence maybe you problematic area.