Two birds, one stone: Add a 5th umpire to the standard crew rotation who serves as the replay administrator and official scorer for the game.
On whatever plays are appropriate, he'd have immediate access to all video feeds and could provide a booth reviewed response via an earpiece to the home plate umpire upon request. The umpire would not have to leave the field, could be resolved in less than 30 seconds, and the process would be completely invisible to the fans.
The same ump would also serve as an independent arbiter for scoring decisions, eliminating the bias which often is driven by the stats of the home team.
The position would be part of the rotation umpires currently have. It adds another 15-16 full time jobs for the umpires (pleasing the union) and resolves a number of problems. Seems like there's no downside. Am I missing anything?