Sydney strategy and Cockatoo Island cruiser building

Would the Sydney strategy's greater cruiser demand improve the chance of ships being built at Cockatoo Island?

Yes, some cruiser construction there is plausible, though the likely number is uncertain.

A Sydney-based Far East strategy would increase the number of cruisers required to support the fleet. That larger demand creates a plausible opening for Australian construction rather than assigning every hull to British yards. Some Cockatoo Island-built cruisers are possible, although a large or precisely defined Australian share is unsupported.