Is it permissible to transfer a sum of money designated for a well-off woman of ill conduct to another deserving person, given that the original donor insists on giving the money to that woman?

Accusing people of immoral acts is a serious matter and is not permissible. Allah has laid down conditions for accusing someone of adultery, among which is the certain witnessing by four witnesses.

As for giving money to that family, it is not permissible to violate the order of the one who entrusted you. However, if you are certain or strongly suspect that they will use it for disobedience to Allah, then it is not permissible to deliver it. You should advise the giver of charity, and if he does not heed the advice, return the money to him and refuse the entrustment.