Can I file a claim against the contractor's license bond in addition to suing him?

Yes, and pursuing both simultaneously, rather than choosing one over the other, is generally the right approach. The standard California contractor's license bond, typically $25,000 for most classifications, covers defective or substandard workmanship in essentially the same way it covers a contractor simply abandoning a job entirely, both are valid grounds for a bond claim against the surety company that issued it.

A bond claim against the surety is typically both faster and less expensive to resolve than a full lawsuit, since it's a direct claim against an insurance-like financial instrument rather than a contested court proceeding. Importantly, filing and pursuing a bond claim doesn't prevent you from also pursuing your lawsuit against the contractor directly, the two paths can and often should run in parallel, since a bond claim alone may not fully cover damages that exceed the bond's cap, especially where other claimants are also filing against the same limited pool of money.