Ruling on meat slaughted by a Kitaabi if at the time of slaughter he says “In the name of the Father, the Son and the Holy Spirit”

It is permissible to eat meat slaughtered by a Kitaabi (a Jew or a Christian) subject to two conditions: 

1.That the meat is slaughtered as the Muslims slaughter it, by cutting the trachea and oesophagus and draining the blood. If he kills it by strangulation, electric shock or drowning in water, then the meat he has slaughtered is not halaal. The same applies if the Muslim does that; the meat he has slaughtered is not halaal.

2.That no name other than the name of Allah, such as the name of the Messiah or any other name, is mentioned over it, because Allah, may He be exalted, says (interpretation of the meaning): “Eat not (O believers) of that (meat) on which Allah’s Name has not been pronounced (at the time of the slaughtering of the animal)” [al-An‘aam 6:121]. And He says concerning haraam foods: “He has forbidden you only the Maytatah (dead animals), and blood, and the flesh of swine, and that which is slaughtered as a sacrifice for others than Allah (or has been slaughtered for idols, etc., on which Allah’s Name has not been mentioned while slaughtering)” [al-Baqarah 2:173].

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: What is meant here is that on which a name other than the name of Allah was mentioned at the time of slaughter, such as saying “In the name of the Messiah” or “in the name of Muhammad” or “in the name of Jibreel” or “in the name of al-Laat” and the like.

End quote from Tafseer Soorat al-Baqarah. 

It says in Fataawa al-Lajnah ad-Daa’imah (22/387): The basic principle with regard to the permissibility of meat slaughtered by the People of the Book is the verse in which Allah, may He be exalted, says (interpretation of the meaning): “The food (slaughtered cattle, eatable animals, etc.) of the people of the Scripture (Jews and Christians) is lawful to you” [al-Maa’idah 5:5]. Ibn ‘Abbaas explained the word “food” here as referring to meat slaughtered by them, which is one of two interpretations of this verse. If a Kitaabi slaughters an animal, if we know that he mentioned the name of Allah over it, it is permissible to eat it, but if we know that he mentioned the name of someone other than Allah over it, then it is not permissible to eat it, because of the general meaning of the words of Allah (interpretation of the meaning): “Eat not (O believers) of that (meat) on which Allahs Name has not been pronounced (at the time of the slaughtering of the animal), for sure it is Fisq (a sin and disobedience of Allah)” [al-An‘aam 6:121], and His words concerning haraam foods: “and that which is slaughtered as a sacrifice for others than Allah (or has been slaughtered for idols, etc., on which Allah’s Name has not been mentioned while slaughtering)” [al-Baqarah 2:173]. If we do not know whether the name of Allah or the name of someone or something else was mentioned over it, then the basic principle is that meat slaughtered by them is permissible.

End quote. 

If it is known that the Kitaabi mentions the name of someone other than Allah, and says “In the name of the Father, the Son and the Holy Spirit,” it is not permissible to eat meat slaughtered by him, because it is something that has been slaughtered for someone other than Allah. It does not make a difference if he means Allah when he says “the Father” – as you say in your question – because by saying “[and] the Son and the Holy Spirit”, it is slaughtered in the name of someone other than Allah. So it is not permissible to eat the meat when these names have been said over it. 

See also the answer to question no. 3261. 

And Allah knows best.