You have 42 days to APPLY for a passport for your child. If your embassy takes time to provide a passport, they should provide you with a letter stating you applied for a passport and the passport will be delivered in XX days. This letter from your embassy will be enough for you to be provided a longer social visit pass by ICA. When you receive your child's passport, you may apply for a LTVP or PR for your child.
Edit: After reading your post again, seems you have actually done this.