Does she want another dog for herself or was she getting a new dog as a companion for Cooper?
If she is doing it for the dog, I would likely have not gotten a new dog in the first place since Cooper seems like he doesn't really want a companion (some dogs are better alone)... IMHO she should NOT keep the new puppy, and based on what you have written here it doesn't seem she will get the help/support she needs to properly train both dogs to get along.
I have raised 2 children and have raised several puppies, the commitment level IMO is pretty similar when it comes to raising well rounded kids/dogs, Does she really want another baby? Does she have that amount of TIME to commit? These are questions I would ask her, because if she doesn't want to spend the next 3 years being a mom to a baby/toddler (aka PUPPY) then she needs to do the right thing and either give back the puppy OR re-home it. Puppies are A LOT of work when you are just dealing with the puppy, but add in your other dogs issues and well I just don't think it is going to work out very well.
I'm sorry if that is not what you wanted to hear, honestly my intentions are good (wanting the best for both the puppy and Cooper). I am not going to lie and say I think everything will or could work out either. It isn't like we are talking about something that has a short life expectancy. Labradors have a life expectancy of around 14 years, that is A LONG time. I know 'I" could not handle that type of chaos in my home, nor would I put a dog aggressive dog through that trauma or risk the puppy becoming dog aggressive from a bad experience with Cooper. If Cooper has training issues, then the new puppy will likely inherit them as well, because puppies learn from the older dog in a home, and bad habits are easily passed down. I am being 100% serious when I talk about the new puppy becoming dog aggressive, and honestly that is one of my biggest concerns.
I wish your mom luck with whatever she decides!