I'm sorry to hear that.
Having the discernment to realize one's own limitations and to put the needs and overall happiness/welfare of a pet first is something that a person should not feel ashamed for.
In a last attempt, you could possibly show your mother the threads about your budgies.
If the decision is final, then you can continue to do your very best for both Buddy and Raffa. You can get Buddy a very large cage to better satisfy the need to fly/exercise and to continue the training by using positive reinforcement techniques to hopefully improve the bond and diminish the aggressiveness.
Best of luck with everything.
RIP sweet Tito (Summer 2008 - January 17th 2013).
You are missed and never will be forgotten.