To be honest, I think that Constantine works much better in the DCU than in Vertigo. The whole point of the character was putting him in a room with people like Batman and then standing back and watching the fireworks. He fits there because he doesn't fit there. It's a source of interesting conflict that is entirely lost when you take him out of that context.
I'm not saying that there shouldn't be a Vertigo book, as well. I stopped reading it a bunch of issues ago but I have been a huge fan of it at points. The Ennis, Ellis and Azz runs in particular are all superb. It's a shame that that is coming to an end but, if I'm being brutally honest, the book hasn't been that good in a long time (though Milligan was a big improvement on some of the other writers).
I don't know whether I want to read Constantine War Journal, though. I like Lemire's JLA Dark a lot. He has a great handle on the character. I would love to see him on the JLA proper, too.
"The trouble with fighting for human freedom is that one spends most of one's time defending scoundrels. For it is against scoundrels that oppressive laws are first aimed, and oppression must be stopped at the beginning if it is to be stopped at all."
-H. L. Mencken