The who is an insured section of the CGL form supplies the answer—the coverages of policy A and policy B will not interact in this instance.

If the named insured is designated in the declarations as an individual, he is an insured, but only with respect to the conduct of a business of which he is the sole owner. Furthermore, the last paragraph of the who is an insured section states that no person is an insured with respect to the conduct of any current partnership that is not shown as a named insured in the declarations. So, under policy A, your client has coverage only as an individual for the business of which he is the sole owner; the partnership business is not an insured under policy A.

As for policy B, if the named insured is designated as a partnership, the members and partners of that partnership are insureds with respect to the conduct of the partnership's business. If a lawsuit names both the partnership and each of the partners, policy B will consider all the named defendants as insureds. In this instance, policy B is the one that will handle your client's insurance needs.

Also note that the other insurance clauses on the two policies will not come into play. The other insurance clause refers to “other valid and collectible insurance” that is available to the insured for a loss. The insurance under policy A may be valid but it is not collectible by your client for this claim, so the clause cannot be invoked for this claim.