Alignment is interpreted differently by _so_ many people. It is influenced by culture and religion and personal reflections on what is good and evil. In the context of the game, it is hard to treat it as a hardcore mechanic. C'mon, we are talking about applying something universally in a game that we can't even apply universally in the world. Need an example? Is Capital Punishment Good? Is it Neutral? Is it Evil? No!!!! Move the mouse away from the reply button for now. DO NOT ANSWER THAT! Just think about it. I don't want a political flamewar going on. Just consider the differences and understand that Alignment in a game is not a constant Universal Principle. Attempts to make it one would enforce somebody's interpretation of morality as a game mechanic. I don't want that to be driven by WotC. YMMV
I do not see any problem with a Paladin executing a prisoner. What is all this about them being helpless? If you take a prisoner back to town and they are sentenced to hang, are you going to argue that they are helpless and executing them is evil? Please note - I am not saying that a Paladin should always execute a prisoner. Nor am I saying that the Paladin shouldn't investigate other avenues of problem resolution. I just don't think making a Paladin atone for executing any prisoner because they are helpless makes sense.
I also do not understand why a Paladin can't be a judge. It depends on the character! There is no reason why a Paladin _has_ to defer to a cleric. First of all, a Paladin does not need to worship a god! They can revere the Cause of Righteousness. Even if they do worship, they can still be the best equipped to ppass judgement. Knowledge (Religion) is a class skill. As is Diplomacy and Sense Motive. In my campaigns, a Paladin may very well be capable of passing judgement and administering whatever judgement the Paladin decided was best.
Of course, in my campaigns, I would allow a wide variety of different types of Paladins. Not different alignments, but different motivations. Some might pursue Redemption of Evil, with a big stick in case that doesn't work. Some might pursue Justice. Some might pursue protection of the innocents. Some might wage aggressive wars against Evil. Each Paladin can be a little different depending on personality, affiliation, religious influence and occassionally, circumstances.
If the PC's have time, I would second the option of a trial. Especially with Zone of Truth in affect.
If there is not time, and given the nature these specific PC's, offering the Cleric the opportunity to truly repent would be a nice gesture. Give her the chance to repent and if she refuses, execute her. If she agrees, give her magical incentive to stay true to her path.
For another group of PC's, I see no reason why the Paladin couldn't assure she was evil and then mete out justice. Sure, she was captured by somebody else. That doesn't mean she isn't despicably evil and that she didn't commit crimes against Good. If a mass murderer were captured in a neighboring kingdom and the Lord there offers to transfer the criminal to your kingdom for trial and justice, is the Paladin supposed to argue that it isn't honorable?