PaDandy is correct; the FOPA will indeed protect you, and was designed precisely for this purpose (to protect you when transporting a gun through a state where it's illegal, but it's legal in the originating and destination states). As long as you aren't stopping off at Disneyland for 3 days or something else that could possibly be characterized as other than "driving through", you're fine.
And, I'll point out that, in my experience, the CHP are actually good guys and generally don't mess with folks without reason. Certainly, this isn't true of the local cops in many PRK cities, CHP these days is mostly looking for drunk/unsafe drivers, clearly unsafe cars, and those *significantly* exceeding the speed limit. I rarely drive under 80 MPH (limit is 65 on most freeways) and they don't give me a second glance. As long as you aren't tailgating, cutting people off, or going quite a bit faster than the flow of traffic, you aren't likely to get any negative attention.
-Troy