I can answer this question.

There is an "Action and Interview List" that gives the Bishopric a list of children approaching baptism age, of those "Unbaptized Members" (and that's exactly what it says) and those who are young men who are overdue for Aaronic priesthood ordinations.

The suggested action for those under 18 appears to be "Baptism". The suggested actions for those over 18 is listed as "Baptism" or "Cancel Record".

The name of these who was NOT baptized (only received baby blessing) appear in the ward directory.

So it would appear that the baby would have a record and be considered an "Unbaptized Member" and appear on the rolls of the church.

Then the following information is provided:

A person age 9 or older who has a membership record but has not been baptized and confirmed is not considered a member of record. However, the ward in which the person lives retains the membership record until the person is 18. At that time, if the person chooses not to be baptized despite being given every opportunity, the bishop, with written permission from the stake president, cancels the membership record. However, records of unbaptized members who are considered not accountable because of intellectual disabilities are not canceled. See General Handbook: Serving in The Church of Jesus Christ of Latter-day Saints, 33.6.2.

So it looks like eventually, this record should be cancelled without your efforts being necessary, BUT it would require the efforts of a Bishop somewhere, with written permission from a Stake President.