Recently my players were raiding a criminal headquarters. These were not nice people: drug runners, loan sharks, things like that. However, as the fight went poorly and they were unable to run away, a number of the criminals surrendered. One literally ducking into a cabinet and curling up to hide, even when it was opened.

The paladin proceeded to directly kill one of these surrendered foes, and advocate the rest, reasoning that their involvement with the gang was reason enough. While they were in the criminals base, they were still within the city. This Paladin is also lawful good and a paladin of Bahamut.

They were working with the guard, and have a friend that is Captain of the district they were in, so legally they should be able to clear this up, but I'm trying to decide if there should be a greater consequence. Would it be wrong to have a temporary dampening of his Paladin powers to show his God's displeasure?

For the record, the group is level 5-10 at this point.

Strictly speaking, the paladin didn't do anything wrong. You should do nothing.

You said in commenting on your question that his oath "does not directly have any [tenets] in regards to mercy/redemption. The oath is based around striking down tyrants." Seems like your Paladin does that, and that's enough.

There are many definitions of what is "Good" in the world. IRL when William the Conqueror installed Feudalism in England in the 11th century the moral code of his knights involved three, and only three, virtues: courage, loyalty and generosity. Note that forgiveness is not among these (generosity was interpreted as helping good people in need even when they have no means of repaying the favor). That's just one example of how being "lawful good" can exclude the sort of mercy you were hoping to see.

Your paladin's being lawful good does oblige him to certain codes of conduct, but mercy on defeated thugs isn't part of it, based on your own description of his oath. So he's being true to character. I might have even awarded him Inspiration.

There is a problem, but not with his oath or power.

Whether sworn before a god's altar and the witness of a priest, in a sacred glade before nature spirits and fey beings, or in a moment of desperation and grief with the dead as the only witness, a paladin's oath is a powerful bond. It is a source of power that turns a devout warrior into a blessed champion.

A paladin draws his power from his oath, and doesn't rely on the benevolence of a god. As long as he's staying true to his oath, his paladin abilities wouldn't be affected. And it sounds like the character is an anti-hero who's willing to hurt people to serve the greater good, which is exactly what he did.

...That being said, from a lore and RP perspective, if he's part of a paladin order of Bahamut, they're going to be a bit annoyed with him. Bahamut is:

a deity of good dragons, metallic dragons, wisdom, and enlightened justice (justice tempered with mercy and punishment with forgiveness)

If you want to draw attention to this, it shouldn't be a mechanical punishment, because he hasn't done anything wrong mechanically. Instead, he might get a psychic message from his order, reprimanding him for going too far. Followers of more violent justice gods like Hoar or Tyr might praise his actions and try to lure him to a god more accepting of his willingness to fight.

Keep in mind, this should not be punishment. Again, he did nothing wrong. Any consequences should be purely part of the evolving story, not an attempt to change his character for him.

Don't use in-game mechanics to punish players

It makes them try harder. Either that or they will resent you punishing them. Show respect for their decisions and talk to him about how he is not following his code correctly, rather than coming out of the blue with it and making him angry with you.