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In order for them to prosecute you for insurance fraud
, they have to prove that you intentionally provided false information or hid information in order to get a payment or some other benefit from an insurance company that would otherwise not be payable. Under CA law, this is a felony and punishable by up to 8 years in jail.
In order to prove a case, they have to prove the rug never existed or that the rug was never damaged or that the value of the rug you submitted the claim for was not the correct value. The fact you submitted a fake invoice and they have the business owner is a huge issue here and it is likely that if they have gone this far, they indeed will charge these cases as CA is overly aggressive in the pursuit of insurance fraud.
You should make NO STATEMENTS ABOUT THIS TO ANYONE, none at all. You must get a local criminal
defense attorney which you can find at the same sites used by other attorneys, http://www.hg.org or http://www.lexmundi.com
You need to direct all communications through your attorney and if anyone contacts you about this matter, including your insurance company, you need to not make any statements and refer them to your attorney.
You "bringing charges" (I am presuming you mean suing in civil court
for breach of contract for them not paying your full claim) against the insurance company will not deter the state from pursuing this claim. In fact, it may also make the insurance company more resolved to pursue you with the DA.
You are going to need to deal with this by getting represented by a local criminal defense attorney and you are going to need to maintain your defense regarding the value of the rug and existence of the rug.