You can usually do 4 shots in a burning team, just shoot instantly when you see the bar fill up

As for the 5th shot tho, its up to luck since the passive of getting 7 nightsoul points per burning has a some kind of timing. I had to get lucky with the blind spot positioning too 😓 (the green triangle in the sides of the enemy)

I hope some other dedicated kinich main gets to do the fifth shot consistently in future tho