We don't have sufficient info about the details of your new job to get deeply analytical, I suggest to you that this is the wrong question altogether.
Look, employment in California is at will unless you are represented by a union or have an employment contract. Neither of those is going to be the case if you are working for a law firm. So you are going to be an at will employee and that means that you can be fired the day after you start (or any day thereafter for the duration of your employment) for humming in the elevator, wearing ugly shoes, or any other reason -- or for no reason.
If the circumstances of this new job are such that you are already worried about asserting your legal rights in an adversarial posture with the firm, how long do you think this new job will last? It would take a firm full of attorneys less than a week to figure out a fully defensible articulation of wholly lawful bases for terminating you (or any other employee). In fact, if it is the will of the attorneys in the firm, it will take them just a few minutes to come up with a legally defensible articulation of a reason for rescinding the offer -- if it is a legal offer at all.
I recognize the difficulties of acquiescing in what you think is a violation of your statutory rights. And I think credit checks for employees is a horrible practice. But you have to come to grips with the reality of the negativity that ordinarily results from adversarial negotiations -- especially as a taint taking root in the first few days of new employment.
It would be much sounder for you to have an open and constructive discussion with the hiring person. Why does the firm think a credit check is necessary? What are the responsibilities that cause this request? Give and take, with lots of give on the side that wants the job. Or be prepared for a potential lesson in at will employment.
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Ms. McCall has done a great job laying out the practical implications of taking an adversarial posture against your employer--something employees so often overlook in their quest to understand their rights and remedies under the law.
Given that your employment is likely "at will," it is best to tread lightly. While you cannot be fired for refusing to agree to an illegal credit check, you can be fired for a host of other lawful reasons which a firm full of lawyers will have no trouble inventing as you progress in your employment. I'm not saying it's pointless to assert your rights, but it is important to pick your battles and usually better to work collaboratively with your employer than against them.
This answer is a general interpretation of the law and is not fact specific to your case. Likewise it does not create an attorney-client relationship. You should seek an attorney for a review of your specific facts and documents.
Although I agree you haven't offered a detailed description of the position being offered, the billing aspect of a law firm operation may handle confidential and attorney client privileged information but I doubt you would be exposed to trade secrets as defined in California Civil Code Section 3426.1, So, I tend to agree with you. I suspect they simply aren't aware of the new law and I also doubt the exceptions have been tested in court. Obviously, they can rescind the offer and then you will be looking to hire an attorney. Maybe you get to be the first case cited,,, do you want to be?
THIS IS A GENERAL ANSWER TO A GENERAL QUESTION AND SHOULD NOT BE RELIED UPON AS A FULL LEGAL ANALYSIS OF ANY FACTUAL MATTER. An attorney-client relationship is not established or offered solely as a result of this answer.
It is unclear based on your facts whether the billing position would fall under the trade secrets exception of Labor Code section 1024.5 (AB 22). If your billing position gave you access to fee bills which described the services being performed for clients of the firm, this information is arguably trade secrets. I echo earlier commenters that this is likely an unresolved issue with a new law, so you would need to decide whether you want to be the vanguard on this issue. I also doubt you would find an attorney willing to take a case like this on contingency, given the uniqueness of the issue.