> How would any bag be an obvious diaper bag?

We used to tote around one of these: https://www.ebay.ca/itm/388909924480

Not very fashionable, has wet wipes hanging out the side, the pockets have stitched icons for what they're meant to hold like a soother and diapers, etc.

Having a child as a prop would definitely help, but a frazzled look and some wet wipes hanging out the side would probably help it along quite a bit.