Here's what actually worked for me, after guidance from answers here:
export BASH_VARIABLE="[1,2,3]"
curl http://localhost:8080/path -d "$(cat <<EOF
{
"name": $BASH_VARIABLE,
"something": [
"value1",
"value2",
"value3"
]
}
EOF
)" -H 'Content-Type: application/json'