Registering a trailer in California is no big deal if the paperwork is correct.
1.
Title must be in sellers name and current address.
2.
registration must be current and in the sellers name and address
3. Bill of sale must be in sellers name and address and state amount paid by you with your name and address.
Failure to have proper documentation of the trailer will complicate things and make your life miserable and next to impossible to transfer and operate your trailer.
Hopefully you dont have to deal with the California DMV in a big city. My suggestion to you is when you bring the trailer down stop in some rual DMV and make the transfer. The DMV in rural areas are not as stressed and worn out and I find much more knowledgable and willing to help you get it done.
If your stuck doing DMV in the big city locations join AAA Auto club and do your transfer there.