Inform the immigration office that a new passport application has been made (with a copy of the application or the details showing when the application was made) and that it may not be issued on time before the passport expires.
Should there be any queries as to why your wife no longer has a valid passport, that query will eventuelly land at the immigration office. They will then know the reason why.
This would be a preemptive action on your wifes part, since the residence permit card fulfills the passport requirement when the passport issuing takes longer than expected (§48(2) AufenthG).
An Ausweisersatz (substitute identity document), that is not valid for travel, can be issued in cases where the period is longer than 3 months after the passport has expired.
It also fulfills a major condition to receive a temporary travel document, should one be required (§5(1,2) AufenthV, German only).
§48 - Obligations related to identification papers (AufenthG)
(2) To meet the obligation to have and present identification papers, it is sufficient for a foreigner who neither possesses a passport or passport substitute nor can reasonably be expected to obtain one to carry the certificate confirming a residence title or the suspension of deportation, if this document contains the foreigner’s personal details and a photograph and is marked to indicate that it is a substitute identity document.
(3) Foreigners who do not possess a valid passport or passport substitute are required to cooperate in efforts to obtain the identity paper and ...