Nah
I don’t see why you would do this if you both have moved on.
Like other have said, let the kids do that shyt for them. Maybe help your kids out with purchasing the stuff they need to give to their mom but that would be it.
And to simplify the argument even more, does she buy you gifts for Fathers Day?
If so, return the favor. If not, fukk that. And if both of you are dating it could be disrespectful to both partners.