Oh boy, you have read earlier threads on these subjects right, it has been explained why what you just told us again conflicts with the lore of Fallout 1 and 2.
1. FEV. FEV was West Tek project in cooperation with the Department of Defense. It was originally started as the Pan Immunity Virion project to develop a counter to possible Chinese bio weapon attacks as well as a cure for the New Plague that was spreading through the Mid West.
When it was revealed that it caused mutations that actually increased the strength, resilience, lifespan, and sometimes even intelligence of subjects it was renamed the Forced Evolution Virus.
At some point the FEV research was transferred to a complete dedicated facility: Mariposa Base were the research continued under military oversight.
So why would a Vault also be re-purposed for FEV research when there is already a dedicated facility. Redundancy? Shouldn't the facility be run by the government and the military rather than a Vault Tec oversight administration.
Second, why would the government bring in another corporation to run their FEV experiments?
What is the point of having a Vault were FEV experiments are continued for little or no reason. What was left of the government had evacuated to the Poseidon Oil Rig and would not be monitoring the experiments.
And if they wanted to continue the experiments they had their own facilities, again no need for a Vault. (the Enclave also did not seem that much interested in created mutant super soldiers any more, instead focusing on improving Power Armor. They only wanted FEV in order to create a killer airborne virus. Horrigan was an accident)
Plus it clashes with the premise of the Vault experiments, they were designed for social experiments, not to create armies of possible insane super soldiers.
2. The East Coast Brotherhood. If the Brotherhood sent an expedition Eastwards to find new technology, how did Lyons and his men make it all the way to the East Coast without any support.
The United States after the war is a way more violent and dangerous world than it was during the colonization days several centuries earlier.
The Lewis and Clarke expedition did not completely start from the East Coast and they did have some aid of settlers and friendly tribes along the way.
Post War US is filled with various outlaws, hostile factions, deadly mutations, and lots of radiation hazards, a lot of stuff that can kill people, even BOS members. (especially when the outlaws want their gear)
If Lyons went East for advanced technology, why did he for example not stop at Colorado? Places like NORAD, Denver, and Boulder also contain a lot of Pre War technology, and I am sure that there are people there who need protection.
Lastly. The Brotherhood is not into defending the people of the wasteland. They are a xenophobic order of technology hoarders who isolate themselves from the outside world, believing that it is a matter of time for everyone else to fail and die, and then claim the world in order to rebuild it.
Members who do not agree with this policy leave the Brotherhood or are exiled.
The East Coast BOS are basically the white knights of the wasteland, a faction that was made up because they have such cool looking gear.
The West Coast Elders would never have agreed to allowing Lyons to use BOS gear for his own little war to defend ignorant wastelanders. If he and a limited number of men could make it to the East Coast on foot, a squad of assassins should be able to do so as well.
They would also not approve Maxson's war, especially with all the technology he now possessed. The moment that blimp was finished he would get the order to collect all their gear and make it back to the West Coast post haste.