EBOD - street legalizing a green sticker 2 stroke in CA is exactly the same process as doing a dirt only green sticker four stroke (which I have also done). If your bike has a red sticker or should have a red, it is technically illegal to get aplate. You may be able to find a way (others have), but its not as easy and you will be in a gray area at best.
My bikes were a '90 KDX200 and a '91 KDX250, I know others who have also street legalized KDX's.
Basically, you install the Baja kit (cost about $400 and takes 6 hours or so). Then you haul the bike and a DMV form known as a 'statement of facts' to a state certified lighting inspection station (most repair shops and dealers are state certified). For a small fee ($10 or so) the shop will briefly inspect the bike and sign the statement of facts. If you get the baja kit, the statement of facts is completed in advance and itemizes some of the things that make the bike street legal. In California, this includes having a battery that will allow the tail light to stay lit for awhile (15 minutes?) with the engine off. The shop may or may not test this. BTW, I have never (3 bikes) had the inspection shop check if my knobbies were DOT approved.
Then you bring the signed statement of fact and your registration/pink slip to the DMV (you DO NOT need to bring the bike to the DMV). Most likely the person helping you will not know how to complete the paperwork, so be patient while they figure it out. Not all DMV's are created equal - if you know of a competent DMV, go to that one. You will pay for street registration and they will give you a plate right then. You will have the option of keeping your dirt (green sticker) registration, if you still want to pay for it.
FWIW, your stock KDX lighting coil does not produce enough power to keep the DOT lights on all the time and to keep the battery charged. You can have the stator rewound for more power, or add a seperate on/off switch to the headlight to save the battery power for when you need it.
Hope this helps. It is possible to save a little money designing your own system and assemblig the parts, but CA has a lot of requirments, and when you look at how much time it takes to make your own, the complete Baja kit is not a bad deal.
Dave