The Church has long held that the death penalty is an acceptable option under very specific circumstances. A legitimate authority may execute someone who has been found guilty of very serious crimes beyond a shadow of a reasonable doubt after a fair and just trial if and only if a lesser punishment (such as imprisonment) is not sufficient to nullify them as a threat to society.

Where are you getting these specific circumstances? The Church has historically taught that the death penalty is acceptable in general, I've never heard of your list being attached to it.

As such, the death penalty is not an acceptable option in our modern society for three main reasons:

Our justice system is not particularly just...

Then why is any penalty allowed to be imposed by it? If the death penalty is acceptable, and the civil authority has the power to impose penalties, then it can impose the death penalty. Otherwise all of its penalties are unjust, and it has no right to impose them. What is your infallible method of determining that a country's justice system is just enough to impose penalty x but not y?

Our system of prisons, while riddled with serious problems... is at least pretty good at keeping people locked up these days

What does that have to do with the death penalty? The death penalty being just means that it's acceptable as the proper punishment for some crimes, and that imposing a lesser penalty isn't sufficient for the expiation of the crime committed. Also, prison breaks can and do happen, as do murders and other crimes by prisoners inside of prison, so what is the magic "pretty good" status of prisons that make the death penalty no longer acceptable, and where did you get it from?

Our methods of execution are extremely inhumane. Even if it's licit to execute someone, that doesn't give you carte blanche to torture them, which is never acceptable under any circumstances.

That's your opinion - and torture has been approved by the Church in the past.