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.