Sandy blight to Rabbit Flat

Try flying. No permits or permission from property owners required.

It is all Aboriginal Land. Contact the Central Land Council in Alice Springs.
Roads are there but you have to know where to go and it is pretty unforgiving country if you get lost.
Do not know if you would get permission through there. If you do remember that all that area is dry under the Liquor Act provisions and it can cost you everything including your car if you are caught and there are now Police at all the major communities.
Alternative route is Kintore to Papunya then back to Tanami Road, Left turn then up through Tilmouth Well, Yuendumu to Rabbit Flat.
Cheers
Mark

Notwithstanding the above, there is a good road from the SBJ northeast to Nyrrippi that doesn't appear on the map, but can be seen using Google Earth. The turn off is concealed behind the big reflector on the northern side of the Desert Road at the Sandy Blight Junction. This is where the Len B marker also resides.

An alternative is to go up the Lake Mackay Track to Balgo, then across to RF. The turn off from the Desert Road is at Mt Webb.

It is possible to get permission from the TOs for both options.

The quickest way, though, is probably as Mark suggests

Bob