Dominion battlecruisers and the cost of running them

Were the Dominions offered battlecruisers?

Yes. Australia operated one, while New Zealand contributed to a ship that remained in Royal Navy service because of operating costs.

Australia's battlecruiser illustrates that such ships were offered to the Dominions. New Zealand contributed to HMS New Zealand but could not afford to operate her, so she remained in the Royal Navy. Canada's consideration of Queen Elizabeth-class ships provides a further example of Dominion capital-ship ambitions.