I think you could just change the first line like this:
Code:
<vb:if condition="$show['quickchooser'] AND is_member_of($bbuserinfo, 1, 2, 3)">
where you'd change 1, 2, 3 to the list of group IDs that can see the chooser. If you'd rather list the groups that can't see it, you can change it to
Code:
<vb:if condition="$show['quickchooser'] AND !is_member_of($bbuserinfo, 1, 2, 3)">
This doesn't actually stop a user from using a style, it just takes away the control. I believe any user could still put &styleid=X on the url, if they know the styleid.