Basta atribuir a variável dentro da instrução case:
declare -a ENVIRONMENTS
case $opt in
"Test")
echo "You chose Test"
ENVIRONMENTS+=("test")
;;
"UAT")
echo "You chose UAT"
ENVIRONMENTS+=("uat")
;;
"PreProd")
echo "You chose PreProd"
ENVIRONMENTS+=("preprod")
;;
"Prod")
echo "You chose Prod"
ENVIRONMENTS+=("prod")
;;
"Quit")
break
;;
*) echo invalid option;;
esac
#...
export ENVIRONMENTS