I would rule both Jeddah and Riyadh out. Visas for Saudi are impossible to get without a local sponsor - and are not always guaranteed even if you do have one. From personal experience, even if you did get a visa, neither would be much good for a mileage run as it can take 2 hours plus to clear immigration! By which time you would have missed the return flight.

DOH is easier as you don't need a sponsor, but still need a visa. I think the flight to Doha alsostops in Bahrain out and back, so that might be a better option as you can get a visa on arrival. Also consider Abu Dhabi where you don't need a visa at all. The AUD flight continues on to Muscat giving you a reasonable turnaround time.).

Hope that helps!