Yes, as DJ-CHRIS said, this rat should live because it is these kind of genes that are selected for in evolution, it is the superior rat.
However if you want to kill it, then hand-feed it some poison. The other rats will not eat any poison because once they have associated the smell of the poison with the death of this rat, they will not eat that poison. It is food aversion commonly found in rats, it is how they learn what food is safe to eat.
The "crap flinging" is probably a way of marking territory or showing off to potential mates, whether or not they are present.