I never did find a standard Joomla way of doing this, so I rolled my own function. For the benefit of anyone else wanting to do the same, here it is:
private function subGroupsOfConfigTopGroup($topGroup)
$db = JFactory::getDbo();
$query = $db->getQuery(true);
This function takes a groupID as an argument and returns an array of groupsIDs that are direct subgroups of the original group.
Note: It does not check any further down the group tree. Joomla does allow heavily nested group structures but in my case I know that there won't be any further nested groups below this level so I'm saving processing time by not checking for them. If you're planning to use this function, please be aware of this and compensate for it if necessary.
I hope someone out there finds this helpful. And I'm still interested; if anyone does know of a standard Joomla way of doing this, please do post an answer.