Surely, at the time that the badge was approved for wear, the iconography of the badge was verified and the entitlement to bear such iconongraphy validated? It's a tenuous connection, but still a connection.
Verified by who? Against what standard? The same Militia Department that let regiments choose to "brand" themselves in imitiation of various British Army regiments? The more you start to dig into the origins of such affectations, the more you start to realize that they were, probably more often than not, simply the preference of the Colonel of the day.
Yes, it's a connection - but it's not official lineage. And that is the critical point. I am certainly not suggesting that any Regiment shouldn't maintain a connection to the units that were formed in the same geographical area as a connection to their predecessors, but respecting and honouring those connections are not the same as having an official lineage including them. We should also keep in mind that recognizing those connections based on geography isn't limited just to the War of 1812 units that form the underlying basis of this thread.