Am I wrong to give my mother everything she asks for, despite my wife's objection and my own lack of conviction regarding her spending?

If the mother is not in need of financial support and requests money for a permissible matter, giving it to her is not obligatory but rather an act of kindness (ihsan). There is no sin in doing so, unless it is known that she will spend it on something forbidden. Furthermore, there is no obligation to inform the wife about what is given to the mother, and subtle speech (tawriya) can be used to avoid disagreement. If there are more pressing expenditures, one can respectfully apologize to the mother while continuously showing her dutifulness (birr). The wife should not interfere as long as her rights are preserved.