moonfingers--It'd be my guess that what you wish is rather straightforward : that date written either in Tengwar numerals or 23 August 2012 transcribed into Tengwar
However, you might be a bit more intricate and research which of JRRT's calendars are the most likely to be used by Sindarin Elves, write the day and month part using the Sindarin words and having that transcribed here. Or both!
It'd take more work, but if it were me that'd be part of the fun. Just something to think about!
I worked it out. Your child's birthday would 19 Iavas 2012. Elves prefer to use dates from the beginning of the season rather than using months. But the month date would be 2 Ivanneth.
Whoops! According to the Tolkien Gateway, Iavas begins on 12 August which makes the date 11 Iavas.
According to the Encyclopedia of Arda and five other websites, Iavas begins on 2 August. So the birthdate should be 21 Iavas. I wonder where the Gateway got its dates.
It'd still be 2 Ivanneth.

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