What you call the Lamb/Ivereigh narrative includes the suggestion that Cardinal Burke wilfully misrepresented and disobeyed the Pope, up to and including trying to talk Festing out of resigning at the very end. 
Even so, I see no reason why Burke should "be gone" from the Order of Malta Patronal role. 
It seems entirely plausible that Francis might simply leave Burke in place. Having succeeded in preventing Burke's plans, and having imposed a new 'delegate' alongside Burke, why can't Francis just wait to see what Burke does next? Or just ignore him?
If you believe that a punishment would be expected, hasn't Burke been punished? - by losing this scrap; by losing it so publicly; and by being side-lined even within the KoM?
On the leaks: Parolin has all the information, and so I would expect that it is he (or his department) which is doing the leaking. Some of the leaks must be knowable/checkable by those 'in the know' (i.e., within the Order, and the participants in the investigation).