A name is not a coping strategy someone invented
In the earliest practice around a newborn, the naming is embedded from the start. In the report on aqiqah the transmitted wording that al-Albani preserves is that the child is given a name, and the phrase about smearing with blood is the part he grades out.
And the instruction that the prayer is offered over a miscarried child, with forgiveness and mercy asked for the parents, treats the child as somebody rather than as an event. So does the report that the child is prayed over.
None of that settles the rulings in your school, and lesson three said where those differ. But it settles something else. A parent who names a child that nobody else met is not inventing a private ritual to feel better. They are doing something the tradition has always done.
Qur’an and Sunnah
ٱلَّذِينَ إِذَآ أَصَٰبَتۡهُم مُّصِيبَةࣱ قَالُوٓاْ إِنَّا لِلَّهِ وَإِنَّآ إِلَيۡهِ رَٰجِعُونَ
those who say, when afflicted with a calamity, ‘We belong to God and to Him we shall return.’
Qur’an 2:156
Al-Mughirah bin Shu'bah narrated that:The Prophet said: "The rider is to be behind the funeral (procession), and the walker where he wishes, and the child should be prayed over
Jami' at-Tirmidhi 1031graded sound, albanifull report
Narrated Samurah ibn Jundub: The Prophet (ﷺ) said: A boy is in pledge for his Aqiqah. Sacrifice is made for him on the seventh day, his head is shaved and is smeared with blood. When Qatadah was asked about smearing with blood, how that should be done, he said: When you cut the head (i.e. throat) of the animal (meant for Aqiqah), you may take a few hair of it, place them on its veins, and then place them in the middle of the head of the infant, so that the blood flows on the hair (of the infant) like a threat. Then its head may be washed and shaved off. Abu Dawud said: In narrating the word "is smeared with blood" (yudamma) there is a misunderstanding on the part of Hammam. Abu Dawud said: Hammam has been opposed in narrating the words "is smeared with blood". This is misunderstanding of Hammam. They narrated he word "he is given a name (yusamma) and Hammam narrated it "is smeared with blood" (yudamma). Abu Dawud said: This tradition is not followed
Sunan Abi Dawud 2837graded sound, albanifull report
Narrated Al-Mughirah ibn Shu'bah: (I think that the people of Ziyad informed me that he reported on the authority of the Prophet (ﷺ): A rider should go behind the bier, and those on foot should walk behind it, in front of it, on its right and on its left keeping near it. Prayer should be offered over an abortion and forgiveness and mercy supplicated for its parents
Sunan Abi Dawud 3180graded sound, albanifull report
Translation: M.A.S. Abdel Haleem, The Qur'an, Oxford University Press, 2004.