Praise be to Allah.
The scholars are unanimously agreed that the father has the right to visit and see his children, if they are in the custody of their mother.
In al-Mawsoo‘ah al-Fiqhiyyah al-Kuwaitiyyah (17/317), it says:
Both parents of a child in custody have the right, if they separate, to see and visit the child. This is a matter concerning which there is consensus among the fuqaha’, but they differ concerning some of the details. End quote.
It says in Fataawa al-Lajnah ad-Daa’imah li’l-Buhooth al-‘Ilmiyyah wa’l-Ifta’ (21/205):
If the wife moves out of the husband’s house, or the spouses separate by means of divorce, for example, and they have one child or more between them, then it is not permissible, according to Islamic teachings, for one parent to prevent the other from seeing and visiting their child. So if the child is in his mother’s custody, for example, it is not permissible for her to prevent his father from seeing him and visiting him, because Allah, may He be glorified, has enjoined upholding ties of kinship in the verse in which He, may He be exalted, says (interpretation of the meaning):
“Worship Allah and associate nothing with Him, and to parents do good, and to relatives”
And in the hadith it says: “Whoever separates a mother and her child, Allah will separate him from his loved ones on the Day of Resurrection.”
Permanent Committee for Academic Research and Iftaa’
Bakr Abu Zayd, Saalih al-Fawzaan, ‘Abdullah ibn Ghadyaan. ‘Abd al-‘Azeez ibn ‘Abdillah Aal ash-Shaykh. End quote.
If the mother of the two girls allows their father to visit them, then she has done what is required of her, and she does not have to allow them to stay the night with their father, especially if she fears that that will lead to negative consequences, such as him turning her children against her, or telling them not to stay with their mother, and the like.
But if she does allow it, then it is something that is permissible according to Islamic teachings, because custody is her right but is not an obligation, so she has the right to yield some of her custody rights to the girls’ father.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Custody is a right granted to the one who has custody, and is not an obligation.
Based on that, if the one who has custody wants to give it up to someone else who also has the right to custody, it is permissible for him [or her] to do that.
End quote from ash-Sharh al-Mumti‘ (13/536).
Our advice in a situation such as yours is: if the father is righteous and there is no fear that he may turn the girls against their mother, she should let them stay overnight with him sometimes, so as to ward off disputes and conflict, lest that have a harmful effect on the girls.
And Allah knows best.