"The CA-based company does not have any customers within CA, so the partner's CA K-1 does not show any CA source income (and, my assumption is that the CA partnership has not paid CA tax for the income, since they do not have any CA-based customers)."

I would assume that they are filing a California partnership return, and not paying any tax because partnership income is taxed at the partner level, not the entity level, even in California.    Client should ask the managing partner if a California return was filed, and if it included a California K-1 for him.  He might already have it but didn't think you needed it.