Hi Scott
I live not far from the Tanami Track, and have not long ago been to Alice Springs and back. With the Tanami in the state that it is in at the moment, I would suggest that it would NOT be a good idea to even take a CT across the Tanami, unless you intend to travel at about 20 kph, as the road is absolutely corrugated, and could cause a lot of damage.
On this trip coming back, there was a CT just west of Yuendemu, with the people waiting for an Outback Recovery vehicle, as they had snapped the axle of the CT.
I am no expert, but IO have given you my honest opinion.
Regards
Tjupurula