If injustice is institutionalized, they will probably use the death penalty anyway. If injustice is _not_ institutionalized, then the death penalty is not inherently unjust.
The death penalty is just one more form of justice, where we can rid ourselves of a criminal without housing them for 40+ years. If the justice system is not corrupt, then it is useful. If the justice system is corrupt, then it will happen regardless of anyone's wishes.
As such, the argument that the death penalty is used for injustice, and therefore is unjust itself is logically flawed.