I would speculate that the West coast Brotherhood probally finished the war a good while back in time and are currently busy trying to establish contact with the Midwestern brotherhood even though they have gone rogue. The might want their power armor back. As for Lyons, they might ignore him, or not, they would probally let him rot in the citadel. But if he managed to get into radio contact with them and let them know about liberty prime, then, they might just take over without his consent and reunite the outcasts and eastcoast brotherhood back into the original BOS. A few executions there and a few here and you have a perfect take over. Now, the probem is that where on earth will Lyons find a radio strong enough to contact the west coast, and what if the West coast is hostile towards him. So many possibilites, damn you lore gap. Why must we assume.
If i undersrood right Wbos made lyons an elder after he reported about liberty prime.(from wiki)
Wiki also says that WBOS do respect lyons but because of his
primary mission they would not sent reinforcement and will not contact him. It was hard for them to make such decision.
Also he did tried to contact them( can be revealed from dialogue) but they didnt