Constantine's presence requires an element of magic or supernatural causes, or at the very least a plot that poses a metaphysical threat. He'd have no reason to show up just to go toe-to-toe with Merlyn or Deathstroke as they try to rob the bank or kidnap the President - mundane crime is not his bailiwick. In addition to that, the villain would have to be long lived enough to look reasonably similar in the past of Gotham and the present of the other shows. The best options in that case would be be either Ra's Al Ghul or Vandal Savage, especially as Ra's is already in continuity in Arrow. Both are of course immortals with the patience to execute plans that have working elements in the past of Gotham and the present of Arrow, Flash, and Constantine. Additionally they wouldstill having enough of a supernatural edge to require Constantine's involvement - Ra's is a frequent user of mystical forces, and Savage may very well be Cain. A choice of either immortal would provide a challenge for the Flash, require the stealth of Green Arrow, could likely have a root in Gotham, and would enter the realm of John Constantine. And a team-up of the two immortals might prove epic.
Brandon grew from the awkwardness of his youth into the awkwardness of his adulthood. He is the author of the Eat Your Serial Press title "Ten Years Gone: Pomp and Circumstance" and is a contributor on Maglomaniac, Polite on Society, and What Culture.