I just did this about half an hour ago. It is the worst mini game in the game, but I found something that kind of worked. Got it after about 10 tries.
The birds are semi homing on your location. They also seem to spawn in waves of like 2-3. So what you want to do is wait for 'dead air' when no birds are around, and then get out of the path of the balloons. The birds will spawn after the dead air and will semi home in on you, and then you dodge out of them, and you dodge right into the balloons instead of going straight for them.
You have to get a feel for when the dead air with no birds happens and when the birds spawn in groups and if you can snag a balloon in the dead time then do it, but if not then you have to fake out the birds and dodge into the balloons.
This.
You just have to not charge straight into balloons when the birds are about to spawn.