There's more to the requirement for the exemption than the "support test."
The child is the poster's qualifying child, even if she provides ZERO support.
In order for the partner to be eligible, they would need to pass the qualifying relative rules because theirs no qualified child relationship (son, daughter, stepchild, foster child, sibling, half sibling, or descendents of one of these).
The partner is blocked from claiming the exemption for a qualifying relative because the child is the qualifying child of another tax payer. Even if they are in the same household, even if she provides 100% of the support.
There is no way to "waive" the rights to allow the partner to claim the exemption, even if she wanted to.