There are only two routes going south, the truck route and the "car" route. DO NOT take the truck route, it is really shit. Try to team up with some other 2WD people (even better a 4WD), ask around the other campsites about who is going south.
If the car rteally is too low then take the train, the next one should be on Saturday, the one after that on Monday.
Go to the train station right now and make a reservation. You may have to wait a week or more at this time of year. There is a lot of chaos as the people building the road have absolute priority on getting on first.
If you do attempt the desert take AT LEAST two complete spare wheels and a GOOD pump. Be prepared to let the tyres down and pump them up after short stretches.
If the van is overloaded and underpowered then consider seriously selling your gear. You will be putting the clutch under a lot of stress if you continually get stuck.
Think seriously about what you are about to attempt, the car goes under a lot more stress than you can imagine, it may not make it and it may be better to wait for the train.