Could be a whole bunch of things mate, and I suspect after you've had a chat with a behavioral specialist or vet, they will prescribe a simple treatment where the key will be consistency.
Depending what they think is the issue, part of the fix could be asking your mum to pay attention to your dog each time she pays attention to her dog.
This is a big ask, but given the potential consequence of not doing this would be having an innocent animal put to death, I would think you would have support in helping the dog get over this.
The biggest issue with dogs is that by nature, they want to dominate the pack. You will never break that instinct as it's too heavily ingrained in the DNA of the animal.
This becomes an issue when dogs are domesticated because what would be normal behavior in the wild, is considered inappropriate by us humans. So the key is to teach the dog that their pack order from a dog perspective doesn't affect their standing with humans as such. So giving them equal attention, equal toys, etc would help this.
Therefore, putting her outside when she acts up may actually be causing more tension. Again, the dog may not see her actions so much as being wrong when she has a go at the little dog, but she will definitely see such punishment as favoritism.
Aside from that, I honestly feel that desexing could help. When you introduce the your dog into the current environment she was still a pup and it sounds like the issues started as she matured.
Are the dogs inside? If they are, maybe put yours in a muzzle when she is inside to prevent her potentially injuring the smaller dog as in order to train the two to co-inhabit a space they will need to get used to being near each other AND yboth your mum and you will need to learn to be relaxed in the situation so the dogs don't pick up on and tension.
So at least with a muzzle everyone will be a bit more relaxed.
All of the above is my opinion and my observations having owned several dogs now.
But again, the absolute best advice is to seek professional help.
Dogs don't quite think like humans, so what we may consider logical and obvious is not the same for a dog. Therefore someone who understand dog nature and behavior will be able to present a solution that to us seems dumb but makes absolute sense in a dogs world.
Good luck!
