How to get GitLab CI to rebuild on JIRA issue status change?
I didn't get what's wrong with webhooks
You just create a webhook on Gitlab CI side and provide this URL to your Jira account.
https://gitlab.com/api/v4/projects/PROJECT_ID/ref/REF_NAME/trigger/pipeline?token=TOKEN

PROJECT_ID - unique project id on Gitlab
REF_NAME - your qa / stage / beta branches within repository
TOKEN - uniquely generated access token (gitlab provided)
Look here for more details how to trigger build on Gitlab via this call