Cats and budgies can live in the same house as long as very strict rules are followed, and even then you can NEVER let your guard down.
I've two cats (19 and 8) and two budgies (8 months and almost 2). The budgies live in the living room, and big open concept space, and the cats are free to roam the house as long as my husband and/or I are there. The cage is placed on a table the cats can't get onto and the cats were specifically trained years ago no tables/counters/anything not the floor. The budgies only come out once the cats are locked in the bedroom; both cats MUST be physically seen and given some treats so they are rewarded for staying in the bedroom. If we are not home the cats are in the bedroom. Even when I'm just changing their food dishes, if the cage door opens for ANY reason, the cats MUST be locked up.
I find myself constantly checking and rechecking doors, locks, and windows. Your brother seriously NEEDS to not open your bedroom door. Why is he doing it anyway, it's not his room to open the door to. Put a large sign on your door stating
"WARNING, BUDGIES INSIDE. DO NOT OPEN DOOR OR THEY MIGHT DIE DUE TO THE CAT"
It seems dramatic but with the cat around it's not, it's plain fact. It's nothing against your cat, she's only doing what she was made to do.
Your animals are beautiful and look very snuggly