The ruse would fail instantly you took the camera out of the bag. Or when you have no child, and look like you are wealthy enough to have a camera hobby.
Well, I found one but it brought us full circle. It looks like a grocery bag and says Big Stinky Diaper Bag. https://www.lifesoleil.com/cdn/shop/products/fai1-ecb007-cre...
https://www.marymaxim.com/products/flower-bed-diaper-bag-pat...
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.