Do you have any third party extensions enabled in the backend? If so, please try disabling all of them to see if one of those is interfering.
Also, just to confirm this template has been fully tested in 3.7.2 and Vertex 4.1 and works with no issues, so this is something unique to your setup.
I'm also using SiteGround and attempted the update, but experience the same issue - where the template config screen does not have any of the UI it normally has and there is no option to save changes to complete the update. This seems to be separate from the Vertex issue, but in order to fully fix the Vertex issue, this must be resolved first.
Cloudfare is not being used. SuperCacher was, but I've since disabled and flushed the cache and seen no changes.
Only extensions are a couple text editors (JCE and JCK), RSForm Pro, and JEvents.
It seems like the configuration file is missing from the Joomla update or it is mis-referenced?