While it's true the Midwest brotherhood recruited outsiders, it's 10 years after FO3. Lyon's Brotherhood also seemed to recruit outsiders as well, and 10 years after, I can see this being the stranded operating procedure. Until I see concrete proof of the Midwestern Brotherhood, I'm almost certain it's Lyon's.
It is doubtfully Midwest, but I think it is just as equally doubtful that it is Lyon's. The BoS insignia's you see in Fallout 3 are actually consistent, and their is two of them. Its known to some that you see both variations of the BoS insignia's throughout Fallout 3, however they are consistent, here is why.
The Insignia variation with the cog to the left is on every single last piece of armor of the BoS in Fallout 3, and it doesn't stop their. It is on the captured Vertibird from Broken Steel, and it is on the flag of the Citadel in of the CwBoS. The same variation, cog to
the left of the sword.
Then you can find several area's in Fallout 3 with the cog to the
right of the sword in the insignia but is painted on a pillar or a wall, or on a miscellaneous items, and structures.
The vertibird's from F3 and F4 are I would go as far as to say drastically different in design, and also their is not a BoS insignia on them. Which doesn't really mean anything, but lets move on.
The insignia on the BoS armor in Fallout 4 is the cog to the right of the sword, along with the same insignia on the Prydwen.
I think if it is anything, anything at all from DC it is the Outcast and that is rudimentary to just say "the Outcast". I mean Lyon's is gone, they are history, the "Outcast" are the BoS in DC. If the 2287 is correct, if it has any meaning in Fallout 4, and if Bethesda literally wants to carry everything over from DC to Boston it is the DC BoS, but it's not Lyon's. This is a BoS that has similar if not the same doctrines of the rest of the BoS. They want technology, and they want to expand (like they did in Tactics).