You DO NOT have to ship the firearm through an FFL, to an FFL. The requirement to have a CADOJ firearms approval number applies only to FFLs.
From the CADOJ website FAQ page:
2. I am not an FFL but I want to ship a firearm to a California FFL. Do I have to obtain a Firearms Shipment Approval number before shipping a firearm to California?
No. The requirement to obtain a Firearms Shipment Approval number only applies to holders of valid FFLs.
While I think the whole idea of having to jump through hoops to ship to CA is ridiculous it really wasn't too much of a PITA to get registered with the DOJ and get approval numbers once you were registered. There is no additional cost to an FFL to get registered to ship to CA. Takes a few minutes on their (CADOJ) website.
Now, finding a CA dealer that will accept firearms from a non-licensed person may be the harder part. And CA dealers are notorious for pretty much raping people receiving transfers in fees and such. Plus they have a waiting period.