The truth is The Elders back west left Lyon's in place despite knowing the new course of action he was taking. They simply said if he wished to follow his own agenda he'd have to do so alone. The outcasts left Lyon's without permission from the Elders back west so they are technically traitors. If they went back to California they could be executed for treason. However as the BoS in California is still licking its wounds from its war with the NCR they'd probably welcome the extra manpower. The Outcasts story is kinda sad they left Lyon's to persue the true mission as they saw it to acquire advanced technology. And what happens a couple years later? Elder Lyon's chapter wins a war with the Enclave capturing a hoard of advanced technology and has control the largest most powerful water purifier in the entire world.
They just said don't expect help - even though somebody sent that kid over at least like 10 years after Lyon's got to D.C.
Apparently the brotherhood was suffering internal political conflicts and his mother thought he'd be safer on the otherside of the country. We also don't know when the western brotherhood severed support for Elder Lyon's group. And perhaps Lyon's decision to help the people of the capital wasteland is what caused the political strife to begin with.
I have never seen the Outcast and Lyons BoS attack one another.
Well they can possibly meet one another out in the wastes but its rare, it happens more often after project purity is activated and brotherhood water caravans start roaming in outcast territory. If you wish to really test this find an outcast patrol then dismiss Paladin Cross from the party you should see some action. Veronica does call it a small civil war which could describe the situation they don't actively go after one another but do go toe to toe if they ever meet.
The plot should have been that the two sides went to war with one another. Outcast tried to take the Maxson kid but failed but did manage to steal parts from Optimus Prime. A quest should have been give to get those parts.
Given that the outcasts are in the wasteland looking to hoard technology and Lyon's had rededicated his mission to a more noble purpose. It be fairly oneside, Lyon's would have allies to fallback on The Outcasts by their nature would have no one to turn to for help. Having to retrieve parts for Liberty Prime that the outcasts stole would certainly be interesting.