Assuming no enmity gear:
Provoke yields 1800 VE.
In order for a DD to deal enough damage to overcome 1800 VE, they'd have to generate a total enmity of 1800 (TE). Since TE = CE + VE and since damage dealt deals both CE and VE, we can reverse engineer this.
Colibri are level 45-47, so we'll do this in 3 sets to show how much damage would need to be done to supersede provoke.
x - Damage dealt
Level 45: x*2.424 CE + x*7.27 VE = 9.694*x TE
Level 46: x*2.353 CE + x*7.06 VE = 9.413*x TE
Level 47: x*2.286 CE + x*6.86 VE = 9.146*x TE
Provoke < Damage Dealt --> How much damage needs to be dealt to equal provoke.
Level 45: 1800 < 9.694*x --> 185.68 --> 186 damage
Level 46: 1800 < 9.413*x --> 191.22 --> 192 damage
Level 47: 1800 < 9.146*x --> 196.81 --> 197 damage
Note that a LOT of that damage will be done in VE which decreases at 30 VE a second, so that unless the puller is right at camp and the mob is provoked right away, even one second will place the above damage below the enmity of provoke. This also assumes that the PLD or the WAR is not using any enmity gear (which at 37-40, they probably aren't).
However, there's one more catch. The initial puller generates 200 VE in addition to the action they used to pull. So, this lowers the threshold to
Level 45: 1800 < 9.694*x + 200 --> 165.05 --> 166 damage
Level 46: 1800 < 9.413*x + 200 --> 169.98 --> 170 damage
Level 47: 1800 < 9.146*x + 200 --> 174.94 --> 175 damage
In other words, at that level, I really find it hard to believe that the damage from a gun wielding COR or RNG dealt sufficient damage to override provoke with a single shot. A second or third shot? Maybe, but not from the initial pull. And, the tank should have done something besides just provoking by that point. All CE and VE numbers used (Kanican's Enmity Tables. No variables were harmed in the manufacturing of this post.