I agree with what @scylla said but would state MUCH more strongly: the fact that the child lives in Canada (even when a citizen) does NOT mean that the mother will be able to come and live with him.The mother agree to give the full custody. Knowing that in the long run as the biological mother she will come here in Canada and live with him. Can the child petition his biological mother in the future…?
The mother should not do this on that basis or understanding - she needs to understand that it does not even mean that she will be granted visas to come and visit, let alone stay. (While IRCC won't maliciously deny a visitor's visa, they can also decide a child of 14+ years can go visit her)
For the child - now 13 - to sponsor the parent, under current rules that is at least 10-15 years away (assuming the child wants to and makes enough money to sponsor her, for several years in a row), AND is drawn in the parental lottery.
That's under current rules and programs for sponsoring parents, which lottery they haven't opened for new applicants now for several years. It's perhaps more likely that they'll change the program (in ways we can't predict) or cancel it entirely.
In short: it would be outright foolish of her to agree to this on the basis she'll somehow get to Canada, and simply cruel to tell her or let her believe that it's the case.
Of course, if she can qualify to get to Canada on her own, that's fine, but then this isn't needed.
And what happens if my wife doesn’t want to agree?
As noted, she can't stop you. That doesn't make it a good idea.