Victorious in the battle for the capital wasteland with the enclave now in ruins and the super-mutant threat largely contained; the east coast Brotherhood of Steel would effectively assume de facto control over of the D.C. area in the aftermath of Fallout 3. Under the stewardship of Elder Lyons the Brotherhood's new mission on the east coast of overseeing the reconstruction of the capital had more to do with the Brotherhood's rather more longstanding traditional goals of seeking to maintain monopoly control over uncovered pre-war technologies than mere noble intentions with the end goal being to re-establish a spiritual successor to the US Army/Pentagon.
While de facto rulers of the capital wasteland and by default the country I don't believe the east coast Brotherhood of Steel's ambitions extended much beyond their techno-religious philosophical leanings, certainly not to the same extent as the Enclave with their grand designs of statehood. Going against conventional wisdom of the Brotherhood I believe Lyon's saw it within the narrow scope of their interest to pacify the citizens of the wasteland rather than risk another protracted and costly war as they had in the core region which would leave them in a much weakened position to defend against encroachment from outside forces and counterproductive to stated longstanding objectives.
So to win over the hearts and minds of the natives project purity was essential, and going forward so would be the clean-up and reconstruction of the infrastructure and the reinstitution of the rule of law. Authoritarian topdown control in the past had given rise to the enclave and nuclear war so I believe would be perceived as neither favorable nor desirable by the public or the Brotherhood and any attempt to initiate any perceived dictatorship would probably met with anger, resentment and violent hostility and the breakdown of relations, not to mention the Brotherhood's complete lack of political aspirations and wider residual sense of detachment and isolation from the local irradiated inhabitants.
In this regard I believe locals will be permitted near-total independence over their own political and economic affairs (where those interests don't interfere with BoS's) and the reinstitution of democratic institutions will largely be handed over with impunity to Rivet City's local academic community with wider participation from wastelanders enforced by Brotherhood and Rivet City join security forces. In this way the Brotherhood will be viewed as liberators and supporters of democracy, even at times with an uneven and heavy hand.
The capital preservation society as the upmost authorities on u.s. constitutional law would be likely candidates to play a key founding role in the framing of the constitution of a new District of Columbia republic and would probably go on to form the basis of the new republic's supreme court. Further amendments would likely grant equal rights to all citizens (e.g. ghouls), freedom of religion (e.g. church of atom), and emancipate all slaves (e.g. paradise falls) drawing direct inspiration from the original founding documents themselves, e.g. federalist papers, etc. For as long as these conditions of relative freedom, liberty and security endure the new republic will likely to grow to become a regional power attracting both immigrants and neighboring hostility. Economic power would probably preclude greater territorial expansion of the Brotherhood of Steel who would benefit enormously from mandatory state taxes and subsidies, etc.
The east coast Brotherhood by this time will have amassed an enormous amount of power easily comparable to that of any of the core region's major powers and within 50-100 years would probably look outward and start conquering new lands whilst simultaneously realizing the importance of the D.C. region's continued financial backing (and hence it's protection) to their present military industrial capabilities having now started developing technology of their own. Economic development will have rapidly expedited D.C.'s transition by this time into a thriving cultural metropolis and the stigma of once being the capital wasteland will be well and truly gone. The west coast Brotherhood are now are a distant poor relation.
Enter Fallout 4. The Brotherhood's northeastward expansion into the former northeastern united states territories. Here they will meet old foes and new ones alike. The Commonwealth is now overrun by the android rebellions while the Enclave are now forging uneasy alliances with unlikely bedfellows (feeling threatened by the sudden emergence and strength of the EBoS) along with numerous other new contenders on the regional scene are now all vying for control of the former New York state. The Brotherhood leadership at this point are no longer content with merely restoring pre-great-war relics now that they have the acquired the necessary technological capacity to create their own new superior technologies and henceforth are now focussed on the new primary goal of becoming the region's single dominant military power thereby rightfully securing their claim/legitimacy as the United States Army.
Under such a scenario the Brotherhood's leading role could potentially be reversed from Fallout 3 as the BoS could rightfully be viewed in Fallout 4 under such a scenario in a different light; in some case as an evil hegemonic invading army who threaten the existence of many newly formed small independent states - blurring the line between good and evil. Or as a misunderstood bunch of crusaders who want to liberate these peoples from the dangers of the Enclave and beyond, or a bunch of thieves/hired mercenaries attempting to steal the resources of the indigenous peoples by force at the behest of big business in the capital - in some cases propping up cruel regimes, etc. In such an event you'd surly be able to take sides and a number of potential outcomes could ensue.
A) The Brotherhood of Steel could fail to secure domination of the region and major rivals could threaten to engulf the entire region in another bloody civil war potentially risking yet more nuclear annihilation.You may end up determining the players.
B ) East Coast Vs. West Coast. The Brotherhood prevail and become the single-unified-military-fighting-force in the east, representing a major threat to the core regions major powers and strategic interests and a power struggle ensues - the midwest becomes an unstable power vacuum as the west unites in the face of any perceived challenge. It would be hilarious if they were all commies!
C) The Brotherhood could lose the region entirely as another power gains primacy and may be forced to retreat all the way back home to Lost Hills humiliated as losers.
D) Stalemate in the northeast = New Miami sunshine! The BoS find themselves in a position whereby they must expand quickly southwards in order to consolidate enough power to counteract growing momentum of a perceived threat of Northern aggression quietly encroaching on their territorial boarders. Eventually a more powerful BoS might then be able to divide and conquer the entire region followed by B ).
E) The Brotherhood's respective splintering into different sects separated by vast distances and the subsequent divergence from their original mission precipitates an eventual civil war, with outcasts, midwesterners, east and west cost brotherhood factions all taking sides.
F) A future without the Brotherhood playing any kind of significantly over-arching role.
G) You decide.