As for canon ending, fallout goes with the good ending so the BoS take over the calculator and they let mutations join. The ending I like can also be a good ending for all but mutations. It would be a story of betrayal and a struggle mutant rights. Mutations were let into the BoS and treated like brothers. They helped the MWBoS win the war against the calculator and in the end they get sent to work/deathcamps. It would be the first time we would see a ghoul/super mutant faction along with humans fighting together for their own lands/and right to live. If they make it into another Fallout Tactics game it could be the first in which we can play as a mutation.
Anyways debating Tactics is not what this thread is about.
Your trying to establish the reasoning for the Lyons being sent to DC and brought up the MWBoS. You said you don't believe why they wouldn't be involved. You opened the door don't complain when i walk in. I'm simply stating that the game was largely disowned by the original creators of Fallout so its not trust worthy. That part aside
You have no basis to support that it follows the good ending. Fallout 1 and 2 require going with the good ending as the alternative is the end of the world. Fallout appears to go with the "Good karma" hero. The statue in Fallout 2 shows the hero of the original was good karma.
And the fact Marcus mentions traveling with the Chosen One in NV and calls him friend implies good karma there to. So Fallout goes with good karma not necessarily the best ending, Which means we can only say the Warrior in Tactics was a good person. As its stated there number is small and they never reestablished contact with Lost Hills the canon ending must be the Calculator is gone as its the one that fits best.
Now back to other subjects why was Elder Lyon's sent out east?
Because the BoS knew there would be rich and untapped caches of technology there. And some travelers from the east brought tales of super mutants, hence the second reason for going. Say someone from DC goes to Pittsburgh with tales of super mutants, someone from Pittsburgh brings the story to Indianapolis, then to St. Louis with another merchant and then onto Kansas City, then to Denver and Salt Lake City and so on. The reports are from travelers. Lastly they still had no contact with the Chicago detachment[once again implying the calculator melted ending].
The war between the NCR and the BoS is first mentioned in Van Buren[which was never published]. We didn't know the war was canon until Fallout New Vegas. Most importantly we don't know when the war started simply that it was after 2242[when FO2 occurs]. It could have been 2257 or 2265. Well after Lyons was gone and to late to call him back. Especially if the vehicles they used to cross the country no longer worked. As the war isn't mentioned in Fallout 3 we should assume it didn't begin until after they lost contact with the west coast.
We don't know the whole details of the journey they may have taken airships and simply abandoned them when they no longer functioned at some point along the way. The loss of contact with the Chicago detachment could also explain the armor. The Elders didn't want to risk losing that valuable armor so they sent them with only T-45b's hoping they'd get better armor in DC[and they didn't] Ashur for instance is back in Pittsburg and has his old T-45b suit implying those were sent with Lyons. *The real reason is story purpose, Bethesda wanted to make the iconic power armor special to honor the series*