Well, I don't see much merit in this reasoning because it is based on a false premise that there is actually any significant difference between mercenaries and bounty hunters. Both are limited to fulfilling bounty contracts. The only difference lies in mercenaries being able to fulfil also unlawful contracts. That is why the ID itself is only the issue because of technical limitations.
There is actually quite a significant difference between bounty hunters and mercenaries. Bounty hunters hunt bounties... Obviously. Mercenaries on the other hand are hired soldiers, often used in lieu of an actual military force (the sort of thing non-house groups would rely on I'd imagine).
EDGE strikes me as more of a PMC than a bounty hunter organisation, and saying that hunting bounties is not your main goal, rather providing military assistance to groups like the IMG, pretty much makes you mercenaries, not bounty hunters, which makes the Mercenary ID, IMO, the most appropriate.
And with regards to the ID, nothing says you cannot be hired as a group to provide military aid. For example, the IMG could pay you 100 mil to kill any Outcasts you find in t23, then I believe you're legitimately permitted to go and kill any Outcasts you find in t23 in compliance with the contract. The Mandalorians have been doing stuff like that for ages.