Both the police and the license board have the authority to investigate a miscreant contractor. But whether they have the will to do so is within the discretion of each. Apparently, they each want to pass the buck to the other agency. You can try to work up the chain of command to get someone in authority to place pressure on an assigned investigator to take the case more seriously or you can take matters in your own hands and sue the contractor. Do you want your money back or do you want to see the contractor criminally prosecuted?
They say you get what you pay for, and this response is free, so take it for what it is worth. This is my opinion based on very limited information. My opinion should not be taken as legal advice. For true advice, we would require a confidential consultation where I would ask you questions and get your complete story. This is a public forum, so remember, nothing here is confidential. Nor am I your attorney. I do not know who you are and you have not hired me to provide any legal service. To do so would require us to meet and sign written retainer agreement. My responses are intended for general information only.
They each have some jurisdiction over this, but I would anticipate that the police will say this is really a civil dispute, not a criminal matter. The CSLB will investigate the claim and possibly send out an industry expert, and then, depending on what is found, refer the parties to arbitration - or if there is reason to believe there is a violation of contractors licensing law, maybe issue a citation to the contractor.
Your best results might come from a civil lawsuit against the contractor.
Talk to a local construction attorney.
It is very common for law enforcement to try to deflect responsibility for a matter to civil courts or to other government agencies. It seems that a big part of police work, aside from writing tickets, is saying "not my job." What you should really want here is your money back first, and getting some contractor prosecuted isn't going to help you in that regard. I would recommend seeking reimbursement civilly, with the help of a lawyer, and then deciding afterward if you wish to pursue other avenues. Never threaten government action or criminal prosecution to get your money back as that is illegal and in itself a crime. I strongly recommend you hire a lawyer.