All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
What you did is considered a lie and changing the truth, and both matters are forbidden. For more benefit on the prohibition of lying and forgery, please refer to Fataawa 119115 and 88412. Therefore, you did well by repenting to Allaah, but this repentance should meet the conditions of a sincere repentance which we mentioned in Fatwa 86527.
However, since changing the date of birth of your son has become impossible, then do not worry about changing it, it is sufficient for you to regret what you did as Allaah does not burden a soul more than it can bear. Moreover, you should think good of your Lord as He accepts the repentance of those who repent because despairing of the mercy of Allaah is a great major sin. Allaah Says (what means): {“And who despairs of the mercy of his Lord except for those astray?”}[Quran 15:56]
For more benefit, please refer to Fatwa 92083.
Allaah Knows best.