You won’t get white number plate self-drive car from any good agencies, since it is illegal to give private vehicles (ones with white number plate) on rent.
Furthermore, the taxi union has check posts en route to Khardung La and Pangong Tso to check the papers of the vehicle. If the vehicle isn’t in your name or that of your immediate relative, they will turn you back.
So a better bet would be to hire a self-drive vehicle, drive it till Leh. Rent a motorcycle or taxi from there to Nubra Valley and Pangong Tso.