If the division weren't dispatched to the adjacent coast, I don't believe that they would not have ideologically reformed to become a military fraternity, at least not to the relative potency that the organisation exhibited in the current timeline. Not to mention how technology from the East would not be extracted. The Enclave, then, would have the ability to insert their monopoly upon the DC Wasteland, where no other faction could pose a political threat to the organisation. Equally, with no presence of the BoS on the East Coast, the Institute, too, would gradually exert and exercise their jurisdiction on the Commonwealth - rendering the Minutemen and Railroad, two puny and concealed organisations, redundant.
With more divisions situated in the west coast, the battle between the Brotherhood of Steel and the New California Republic would, in my opinion, still favour the latter, due to how they are a nation that can sustain itself during wars, compared with the introverted, isolationist apparatus of the former. However, I could envision the BoS-NCR war partially stagnating, with the divisions, which are equipped with power armour and effective weaponry, to halt NCR advancement in certain strategic locations. In this timeline, I still concur to the sentiment that the Mojave Chapter would be catastrophically defeated at Helios One: Elijah's hubris and blinded obsession over the location stationed the BoS in a exposed and vulnerable position. Alternatively, it could be argued that the BoS could have occupied the Dam with the enlarged force - however, that would be solely dependant on the idiosyncrasy of the Elder.
Subsequently, this alternative timeline would witness the East Coast become engulfed by the hegemonic organisations, Enclave and Institute. Meanwhile, the West Coast, as expected, would bolster the manpower of the conglomerate - resulting in adjustments to sociopolitical events, chiefly revolving around the conflict between the NCR and BoS, which I predict would eventually lead to stagnation or a NCR victory with higher calamities.