Wow those screams! lol Ive never had a macaw so im not sure if what i suggest would even work but if it was me, I try the following things:
1. Give him a new interesting toy before handling another bird. Maybe if he is distracted with a new toy he would be happy to entertain himself for awhile.
2. I would give him a bath before handling another bird. Usually a bird spends quite a bit of time preening after the bath and hopefully that would keep him busy long enough while you interact with the other birds.
3. I would only try this if the other two failed...I would use a clicker and his favorite treat. When you handle another bird i would count out loud " 1 missassippi, 2 missassippi, 3 missassippi, etc." Click and treat after 1 second of him being quiet while you are with another bird, then after he has mastered that, increase to 2 seconds and click and treat if hes quiet for 2 seconds while you interact with another bird...etc. keep going so hopefully he will understand him being quiet while you interact with another bird will earn him treats.
I hope one of those things work. Im sure your desperate to find a solution!