Can I, the client, use the threat of a criminal proceeding to help settle a civil matter?

You're right, in California a lawyer may not threaten criminal or administrative action to gain an advantage in a civil suit, but you as long as you, the client and non-lawyer, are under no similar State Bar regulation, I suppose you could exactly that. If you are represented by a lawyer, you need to be careful that you don't implicate your lawyer in this threat.

Another thing to consider is the public nature of criminal claims. You can settle your own civil claim, but this criminal action would be prosecuted in the name of the people, and as a matter of public policy, you cannot agree, for example, to refuse to cooperate and tell the truth if you're subpoenaed by a prosecutor over this criminal matter.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.