If a California resident buys a business in Wisconsin that is held in a Wisconsin LLC, do they need to file a fictitious name statement for that LLC?

Your question is not clear, and it is further unclear how and why a fictitious name is being used at all. If the buyer has bought the membership interests of the LLC, the answer might be different from that in a circumstance where the buyer bought the assets of the LLC. You also don't specify where or whether the LLC (or the buyer) is doing business under a name different from its own. Since  you don't provide this information, your question can't be answered. See an attorney and have him review the specifics of the transaction to advise you properly.