Hook left from Melbourne to Port Hedland and then to Derby +/- Wyndham.
With a brute force approach focused on dumping several hundred thousand tons of supplies + four or five additional divisions of troops ( which you DO have available within Australian Divisions ) either landed by ship or just marched up from south-eastern Oz you can take Darwin. I've had success in similar situations by just dumping huge gobs of supply at Wyndham etc and lettting it trickle into Darwin.
Of course, if you're really ballsy you could just ignore Darwin once you take Derby and the base to Darwin's north, put 300 fighters on CAP there and dare the Japanese into a force on force fight with you. From that base you can exploit north and bomb and bombard Darwin until it runs out of supply and then take it at your leisure. You never see people talking about taking that base but in one game I did just that and cut off ten divisions in the Australian hinterland ( after luring them there with a show of weakness

). That base can be built up to a very respectable AF size. I put BBs and CAs etc at it to defend it, shredded his air force whenever it tried to take them out and traded him BB for BB and CA for CA when he came bombarding.
Eventually he brought his CVs into action to really reduce it. I fought him tooth and nail, losing about 300 or 400 planes in return for gutting KB's flight wings ( killed about 300 of them ) and then unleashed my invasion of Marcus/Wake and the Aleutians a week after he'd just gutted KB in a sideshow. A week after that my opponent resigned. Your opponents won't resign but there's no absolute need to take Darwin, you can use it as a potential fulcrum to draw him in, get him to commit KB and then allow you to unleash your blue ocean offensives without fear of KB.
Really, if you look at the position you don't NEED Darwin. You need A base in that region from which to springboard.... it being Darwin would be nice but not essential. Hell, if you were all maskirovka-y and sneaky it is actually better for that base not to be Darwin since that plays into a narrative whereby a counter-invasion ( and commitment of the reserves you want to be out of place when you hit Marcus and Georgetown ) is possible --- and indeed, subtly, encouraged by you.
The easiest way to lead people is to make them believe that there's an opportunity and that they are following their agenda. Manage their concept of what is and isn't achievable and, assuming you understand their motivations, they'll happily do whatever it is you wished them to do under their own steam, wasting their initiative to do precisely what you wished them to do.... opening up opportunities for you to do what you wish unto them in other theatres.