Shared on the studio's Instagram account, the cartoon image depicts a cross-eyed Harry, a stressed Meghan and a screaming Archie.
Many followers instantly took offence.
"I do not like this illustration of the Royal Sussex Family," while another slammed the artists, "This picture is trash!"
Another insisted, "The image is not polite, nice or gentle. It is disrespectful."
The mock-up coincided with the arrival and release of the first images of little Archie, who posed in the arms of his deliriously happy parents on the grounds of Windsor Castle.
Meghan, who was glowing in a white dress just two days after giving birth, told the world she couldn't be happier to be a mother.
"He’s just been the dream so it’s been a special couple of days," the former actress gushed, before praising her husband.
"It’s magic, it’s pretty amazing. I have the two best guys in the world so I’m really happy."