I figured out an extremely easy way to kill off a Big Daddy.
I just use the security bullseye plasmid and toss it onto a Big Daddy, and I intentionally activate a security camera.
The virtually endless waves of flying security bots are very effective in killing a Big Daddy. Usually, I only need to trip the security camera once, and the Big Daddy falls down in less than 30 seconds.
I use a single dose of plasmid, and the Big Daddy is dead.
Perhaps it's a glitch in the gameplay. It's just too easy.