The investment banking market in Singapore isn't doing so great at the moment. Deal activity is low, and there have been massive lay-offs. Rather than competing against experienced hires, you may have a better shot positioning for an undergrad role. You will not be considered an experienced hire since, well, your back office experience will be (fully) discounted.

Network extensively within the bank to see if there are any openings. I've seen product controllers move to sales/trading or trading from BO if the trader likes you, but I have never seen a direct move from back office to investment banking (not unless you go via the MBA route). Obviously happy to be corrected here.

The fact that you have a Master's degree from Australia, and only being able to land a back office role on the outset, may point to the fact that your competitors (other grads) have more impressive resumes and/or relevant experience. I still wouldn't rule out going for a top MBA programme, since they tend to be regarded more highly.

Happy for others to chime in here, but I don't like your chances given your current position (i.e. no top MBA) and I doubt accumulating more experience in back office will help. Maybe you can consider a corporate banking role which still pays better than back office.