Thanks so much for detailed answer sirUpdated. Congrats!!
You both will have to do Oath same time in the ceremony room. You can take the baby in the ceremony room. As long as baby is not crying a lot, you should be ok. Even if baby is crying, one person can take baby out of room. They will announce that. And during oath, just bring the baby and have Oath in 10 minutes (including signatures and receiving certificate). People sometimes give baby 0.5 ml of liquid Tylenol and that makes baby sleepy. It is highly recommended if travelling by Air on long flights with babies. I gave Tylenol to my baby during travel to India. No harm and sometimes doctors also say you can give some Tylenol to baby if baby has cold/ cough and Tylenol makes baby sleep.
Note: Do NOT make it a regular practice. But one in a while or so, should not make a difference to baby's health.
Appreciate it