It depends on own preference, some people apply for it as it never expires, some people apply for it as they don’t want to apply for or renew passports unless required. If i were you, i wouldn’t waste ~$1200 on that.

Regarding evidence of your child living with you, you don’t need to worry about it. Your child, when in the US, is self understood to be living with you, unless it’s a divorce case.

U.S. Citizenship through Naturalization of a Parent

If you were born outside the United States and acquired U.S. citizenship through the naturalization of your parent(s), please submit the following with your passport application:

Your foreign birth certificate listing your parent(s)
Your parent(s)’ naturalization certificate
Evidence of your permanent residence status. Examples include:
Permanent Resident Card/Green Card
Foreign passport with the original I-551 visa entry stamp
Your parents' marriage certificate (if your parents were married when you legally entered the U.S. and before your 18th birthday)
Documentation of legal custody (if your parents were not married when you legally entered the U.S.)
Evidence of your legitimation (if your parents were not married at the time of your birth). Examples include:
Your parents' marriage certificate dated after your birth
Certified court order of legitimation