It's not like they have veto power.
The Diamond is run by an authority that has a lease with the Braves for the AAA club. Anyone's guess how it'll all get structured if and when they renovate or move the stadium. At the rate we're going, there's talk we may lose the R-Braves, at which point we'll be hungry for a major baseball event.
VCU plays at the Diamond and talks from time to time about buying it if the R-Braves pull out, but UR has its own playing field on its campus. That field could not accommodate a major tourney -- not enough stands or other facilities.