What was/were Madame John's original motive(s) for not telling Tite Poulette of her background, and would Mme. John face any consequences if people were to learn that Tite Poulette is not truly her child by birth in "Tite Poulette"?

Madame John's original motive for not telling Tite Poulette of her parentage is twofold: the child is the orphan of Spaniards and would be taken from her because she is a quadroon. By keeping the child as her own she has, perhaps, made Monsieur John feel better because he knows the girl will be well cared for since he has put her under his legal protection as his child. 
At the time of the setting of this story, there was an Orphan's Asylum where the beautiful Tite Poulette would probably have been taken if Mme. John had not claimed the child as her own. Because of his guilt about having given the baby's father coffee and toast which actually caused the death of the man suffering with yellow fever, Monsieur John felt responsible for the baby girl. Since he was not married and obviously loved Zalli (Mme. John), he left her his house. As he is dying, he tells the woman he loves that he is going and he recognizes how faithful she has been to him. He also remarks that with his passing, there will be no one to take care of her.

Zalli only went on weeping."I want to give you this house, Zalli; it is for you and the little one."

Monsieur John then signs over his property to Zalli before he dies. He also makes a "sworn statement" (alluded to at the end of the story as Mme. John reveals the truth about her daughter to the physician), which apparently indicates that the baby is his.
If people at the time of the setting of this story knew the baby was the child of the Spanish couple, the child would probably be taken from Mme. John, who is not white. This would be done because of the separation of races. For one thing, the girl would not be permitted into any level of white society if it were known that she lived with a black woman. Besides, no quadroon would have any claim to a white child. It is because Monsieur John knew the mores of his society that he wrote and signed a false statement that the baby girl was his. With such a statement, Zalli would be able to keep the baby if anyone made challenges about her origin.
Madame John's revelation that Tite Poulette is white is, perhaps, given to the Dutchman for two reasons: She realizes that the young man truly loves the beautiful young woman, and she also knows Tite Poulette has little or no future in the white society where they live, because she was raised with her.