I would drop Woomera and spend all your time in the Flinders - its an engrossing place.
Find maps that show Public Access Routes (PAR). These are 4WD tracks that take in phenomenal scenery.
You may be able to book accommodation in Grindel's Hut in the Gammons. It is a full on house at the end of a pound, elevated, with views to die for. And hot showers! Good for the girls to have a break? Organise through parks SA.
When you go to Warraweena (another must!), tell Stony that Wren sent you. I have taken a group to Warraweena doing all sorts of volunteer stuff there for years.......
One way to avoid tyre problems is to deflate and drive a bit slower. In my Discovery, I run 26psi front, 30psi rear; and have never had a flat in the outback, over many, many years of travel. Also stop frequently and walk around the vehicle and look at stuff, including tyres. That way you will have a chance to catch a slow leak and hopefully avoid destruction of a tyre.
Then, next year, you can go to the opal fields and Woomera, take in the Oodnadatta track (more like a really good road than a track) and, and,.........!
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