Answer

My assumption is that such a name was written not for the sake of holiness, both because the man did not intend that, and also because even if he did intend it, he was apparently legally incompetent, and his intention has no halakhic significance.
Regarding erasing a name that was written without intent to sanctify it, see here, section 2 (and the sources are also there):
In Hagahot Maimoniyot it is written: “If he wrote letters of the Name without intending to sanctify them by writing them, they have no holiness.” And so too Beit Yosef wrote (2b) in the name of the Tashbetz: “If he wrote God’s Name without intent of holiness, it is not holy,” and he likewise proved this from the Sefer Mitzvot Katan.
We see that a name that was not sanctified before its writing has no holiness, and it may be erased.
However, in the book Machaneh Ephraim, in his responsum concerning the melting down of coins upon which the Name is written, he raised an apparent contradiction in Maimonides’ words regarding a name written without holiness. On the one hand, Maimonides wrote in law 6 that even a name engraved on utensils may not be erased. And on the face of it, a name on a utensil was not written for the sake of sanctifying the Name. On the other hand, in law 8 it appears from Maimonides’ words that if a heretic wrote a Torah scroll, it is burned, since he does not intend to sanctify the Name, and plainly this is because a name lacking intent to sanctify it is not subject to the prohibition of erasure.
Because of this contradiction, Machaneh Ephraim distinguished between a name whose writer intended it for the sake of Heaven but did not intend to sanctify it, in which case it is forbidden to erase it, and a name written by a heretic who does not believe in the holiness of the Name, and wrote it as though it were an ordinary secular name, in which case it is permitted to erase it.
See also at greater length in Sdei Chemed, who wrote in the name of Mahari Asad: “Even though a name written without specification is forbidden to erase, if it was written not with intent toward the Divine Name, it has no holiness at all and it is permitted to erase it.”
Based on this distinction, the words of the Shakh are explained, as he defined the Jewish law in this matter in three levels:
[1] A name written for the sake of holiness – it is forbidden to erase it even for the sake of correction.
[2] A holy name that was not written for the sake of holiness – it is permitted to erase it for the sake of correction.
[3] A name written as an ordinary secular matter – it is permitted to erase it.