If you turn off the dragon using Birb you will get the 50 Unit bonus and not the 300 Unit bonus for slaying the dragon.
Press the dragons button, note the 300 units given for turning it off.
Disable the dragon with Birb, note the 50 units given for using Birb.
The payouts to stack, meaning that you get 350 as a result of killing the dragon using Birb.
They don’t stack, meaning that you only get 50 units for using Birb.
It’s a fair bit harder to kill the dragon using Birb in a typical round rather than pressing the button, so a reduction in units/exp and it doesn’t count towards the leaderboard means using Birb for anything past the stun seems like a downside. I didn’t test, but I assume this also applies to the Mystic Grove sword due to how it relates to Birb normally.