We appear to have it figured out. This is really weird…
Our JIRA instance had webproxy parameters specified in its java startup:
-Dhttp.proxyHost=webproxy.ourdomain.com -Dhttp.proxyPort=80 -Dhttp.nonProxyHosts='135.*|*.ourdomain.com|localhost' -Dhttps.proxyHost=webproxy.ourdomain.com -Dhttps.proxyPort=80 -Dhttps.nonProxyHosts='135.*|*.ourdomain.com|localhost'
With the webproxy configured, the EXTERNAL traffic to that webhook test site was working fine. Internal traffic, however, was not getting through. We believe we had it configured correctly, though. Our internal applinks between Atlassian applications worked fine.
As nothing else was making sense, we removed these startup parameters. Now, the webhook to Mattermost is working fine.
So, no idea if there was something wrong with our webproxy config, or if the Atlassian webhook engine is just cracked.