It seems that the exportability of a page is dictated by the column "IsPersonalizable" in the table "PageDefinition". The KnowledgeBase page is set to 0 and has no Export button, while others are set to 1 and have an Export button.
Try setting the value for IsPersonalizable = 1 in your test database table like so:
update PageDefinition
set IsPersonalizable = 1
where PageID = '403a6a23-6719-4570-8489-fe02577fce92'
where the PageID is the ID for the KnowledgeBase sub-page.
Then see if that gets you any closer to replacing your busted page. This should at least enable the Export button. That was as far as I tested.
I was unable to find the appropriate web part. It looks as though if you deleted that one, the export/import is the way to go.