Disclaimer: Naruto isn't mine
Being a dad means sneaking a camera phone into the delivery room without your wife knowing it.
Shikamaru didn't consider himself to be a cruel man. He wasn't particularly mean or spiteful. He treated people with courtesy and respect and only occasional annoyance if they interrupted him while sleeping. He never hurt a woman and wouldn't dream of hurting someone pregnant. Female or male. He was the perfect person to spend pregnancy with. So he respected Kiba's wish when Kiba asked/demanded not to take a picture anytime during the labor and until the baby had been born.
Kiba doubled-checked to make sure there was no camera or cell phone in the baby bag. He doubled-checked everyone that was going to the hospital to make sure they had nothing that could be used against him. Apparently, there was a weird Inuzuka ritual about the birthing process. But not taking a picture would be troublesome when Kiba or another member of the Inuzuka family would come around asking to see a picture.
Sneaking one in would be a lot less troublesome.