I was able to get the group members api request to work on meta with this curl command:
curl -i -sS -X GET "https://meta.discourse.org/groups/my-private-group/members.json" \
-H "Content-Type: multipart/form-data;" \
-H "Api-Key: key" \
-H "Api-Username: username"
Maybe you can use the copy as curl command in Postman and compare.