Australian dry docks could support the China Station

Would Australia likely build a dry dock for a larger capital ship?

Yes; Australia already contemplated facilities for HMAS Australia, and expanded infrastructure could also serve China Station refits.

Existing plans to support HMAS Australia make a suitable Australian dry dock likely in this counterfactual. A larger capital ship would require investment in its supporting infrastructure. Singapore would probably remain the Royal Navy's regional hub, with possible additional facilities in Ceylon and elsewhere. Australian yards could nevertheless take China Station ships for major refits. Their crews could spend the refit period resting in Australia.