Here is the law that affects running it is in the California code of regulations, title 14(Natural Resources) section 1.63.

§1.63. Movement of Live Fish.
Except as provided in sections 4.30 and 230, live fin fish taken under the authority of a sport fishing license may not be transported alive from the water where taken.

the exceptions in sections 4.30 does not apply as it only pertains to baitfish. Section 230 refers to the permit process needed for contests(tournaments). Now here is the interesting part, see (2) below. It appears that the DFG does have the power to establish permit conditions governing the movement of live fish associated with tournament activities.

copied from section 230
(d) Compliance with Sport Fishing Regulations.
(1) No provisions of these regulations exempt any participant in a contest from the sport fishing regulations, except that the department, as part of Event Permit conditions, may authorize an exemption to new regulations which impose an increased minimum size limit larger than 12 inches, a slot size limit, or a reduced bag limit less than five fish.
(2) To prevent the movement of live fish from one body of water to another, the department also may establish permit conditions governing the movement of live fish associated with tournament activities.

I think this is the area where we will need to concentrate our efforts. I am still researching this and keep in mind this may not be holy grail we seek since everything I have read so far makes the permits more stringent than the existing regs. not less.