The choices are: 1) submit a PCC with incorrect information or 2) request a new PCC and ask them to correct the error(s). If you go with #1, then it will contradict information in her birth certificate and the immigration forms and guess what? CIC may ask her for a PCC from this "wrong country" she was not even born in. When CIC does background checks, it includes "document verification". A birth certificate from her country of birth will be different from one issued by another country, so yes, CIC may notice and question it.
I would go with #2 - makes life easier. If you are running out of time to meet the deadline, inform the visa office the document requires correction and ask for an extension. They should be able to accomodate your request. Best wishes.