Labor Compliance would not necessarily be responsible for a failure to act. In some instances you may able to get a writ of mandamus which forces an administrative body to act or not act, but you will not get any monetary damages. Your employer however is on the hook for several labor code violations, including but not limited to: failure to pay overtime, waiting time penalties, and incorrect pay stubs, etc. Unfortunately the LWDA is overburdened with a high number of complaints and a low budget. To remedy this, the legislature has given private attorneys the authority to prosecute businesses for penalties, thus if you are serious about punishing your employer you can pursue civil penalties on behalf of all the workers in tandem with your own personal lawsuit. Just remember you only have a year to pursue penalties on behalf of the state.
You cannot successfully sue Labor Compliance to force them to exercise their discretion as to when and how to enforce which wage laws and regulations. And doing so would presumably only be to achieve the eventual remedy of getting properly paid, for which you have a legal remedy - sue the employer.
You can hire a private attorney to get your overtime pay for the company. Be sure to hire a qualified wages and overtime claim attorney or law firm having experience with prevailing wage cases.
I answer questions on Avvo to try to help get you pointed in the right direction. But, I am not your attorney. Beware, my answers here are general, limited, incomplete, and can never be as complete, thorough, or accurate as one I would give to a client after hearing all of the facts and details of my client's situation and applying the correct law. Also, I am admitted to practice law only in California and all of my answers are intended exclusively for the Golden State.
While you cannot sue the Labor Commissioner to force it to take action to enforce your rights, you can file a FREE complaint with the Labor Commissioner to initiate an administrative proceeding as to your claim for overtime only at:
Alternatively, you can seek assistance from an employment lawyer who can sue for overtime and attorney's fees for just you or you and your co-workers as a class action. In addition, you could file a claim under the Private Attorney General Act (PAGA) for overtime pay as well as seek to penalties for multiple Labor Code violations on behalf of you and your fellow employees.
Finally, if you employer takes any retaliatory action against you for making a wage claim, you could also have a claim for wrongful termination or demotion.