I know dogs. Have had them as pets both before and after having children. I also love animals, including dogs, so not a dog hater.
In this case, the family dog, a rottweiler...so a known aggressive breed, has as soon as the mother left the room, bit her 6yr old child. The OP said it was barely a minute because she went to get a package from front door.
Sorry, but it’s a bite because it left a bruise. That’s an injury. Don’t minimise it. And I don’t agree that because it could have been a worse bite than it was, that this is evidence that the dog didn’t mean to bite.
It’s no coincidence that aggressive dogs assert themselves on weaker beings when an alpha has left the room. The dog probably woke when the door bell/knocker went, and was not startled by the child but took the opportunity to assert dominance over the child. The dog is literally telling the child that it ranks higher in the pack than the child does. This is not good. Especially since it appears there will be no consequence, the dog will think it is acceptable to be aggressive towards the child. There will be another bite. It’s just a question of when. The OP is risking that next bite being much worse or a full blown attack.
It’s not a risk I would take. And I’m not advising PTS, but rehoming the dog with a family with no children. I had to do this with a dog of ours when she snapped at one of my young DC. My DC wasn’t even bitten...but I had witnessed the event and saw it for the warming sign it was. I think because OP didn’t see the dog bite her DD, it is out of sight, and out of mind. Less of a problem. The steps taken are not going to be sustainable. Dog and child will be in close proximity again and even if you are there supervising, you cannot move fast enough to prevent another bite.
The breed makes a difference, when looking at statistics, Rottweilers are 10x more likely to maul or kill a child than the average. They are the second deadliest dog breed. I know whether a dog attacks and mauls or kills a child is influenced by environment and type of owners/home but with statistics like that, the observed deadliness of Rottweilers cannot all be pets of bad owners/abusive homes.
As a prior poster said, who talked about the cases she sees of children needing corrective plastic surgery, most commonly the attacking dogs are historically well behaved dogs from loving homes with good owners. Don’t have the “it can’t happen to my child” mentality. It can and this was a clear warning.