We have a very limited space to respond to your question, so I can't cover the entire scope of your question, but let me give you some ideas to put into your thesis.
Kant's argument in favor of capital punishment assumes the entire purpose of the legal system is to administer a narrow definition of "justice". It ignores that another major purpose of the justice system is to protect the public from criminals by separating them from the population. No death penalty is required to do this. Nor does the accused, in any society, get an absolutely equal or even adequate defense.
Kant also, one could argue, confuses the idea of justice and punishment, which are not the same. To achieve justice, wouldn't we have to bring the victim back to life? Otherwise, the loss is permanent and no true justice can be achieved. Thus, public safety is of more paramount importance.
As for deterrance, there is ample statistical evidence that suggests crime rates are the same or higher in states that have the death penalty vs. those that do not.