If the remaining child were to be killed, this too would free the daughter-in-law from her Yevamah status – since the living baby was her only brother-in-law. From the daughter-in-law’s perspective, in fact, killing the child would result in a better solution for her. By just kidnaping the child she might have convinced the earthly court that she was not a Yevamah. However, she herself would know that the child was not really hers and that she really was not permitted to remarry, until Chalitzah was performed. By having the baby killed, though, she would truthfully be released from the bonds of Yibbum.
This is the reason the daughter-in-law suddenly lost interest in keeping the child when she saw that King Solomon was ready to cut the child in half. This would serve her interests even more than if she took the child for herself. Therefore she insisted: “Cut!”