Except, can you? I don't mean like joining the Brotherhood, which is a very clear process etc.
No not really. I'm assuming you know about the FEV plan, but inserting it into the purifer is the only way you can side with the Enclave.
They win if the FEV is inserted, but with Broken Steel loaded, you are still forced to destroy the remnants of the faction you just sided with (stupid, I know).
But do they at least have a central base that I haven't found yet, and is it ever possible to reach a stage in the game when they don't shoot at you?
They have two central bases that they use. Raven Rock intially is the capital and this designation is later given to Adams Air Force Base.
And no, they will always shoot at you, because you will always be supporting the Brotherhood. No matter what.
I presumed this might be the case if you blow up the Citadel at the end of Broken Steel
Nope. Firing on the Brotherhood still causes the Enclave to get blown up as well. Which is ridiculous and its never explained as to why this occurs. Bethesda got lazy.
but I'm not so keen to play all the way through only to find that doesn't work.
You can roleplay as the Enclave, but you can't offically join them without mods installed.