The simple answer is that LA County is a county. California is the state. Unless you are an employee of California (or some other state) you are not eligible,

This does not mean that the property might not offer the rate to you. But, the way to handle that is to call the property directly (not toll-free reservations) and ask whether it will extend the state rate to you. If the answer is yes, that is great. If it is "no" you will know before you get there and can make your decision accordingly.

It is in your interest to keep this all above board. You are traveling on LA County business and the last thing you need is some property complaining to the County that you were flashing an ID trying to obtain a rate to which you were not entitled.