One of the young guys who works for us has been known to play our courses with a putter. His most impressive return was making par on #7 at Streamsong (Blue) from the drop zone ... yes, he hit it 80 yards over water, and then got down in two.
The director of instruction at Bandon, Grant Rogers, is also famous for playing entire rounds with a putter over there. Of course it is not your standard off-the-rack putter ... he has a lot of lead tape on it to hold it together, and he can hit it +/- 200 yards including roll-out.
I think you could get around with a putter at Royal Melbourne. You could certainly do it at St. Andrews Beach.