Dogs in the Vineyard is the first game I think of when asked for an RPG that brings moral choices front and centre. The mechanics push for character growth, and moral issues is something the GM is pushed towards to challenge the players.
Dogs in the Vineyard is all about asking moral questions. The default setting of the game is something like this: Out just barely in the settled regions of a slightly fantastic American Wild West, a religious order has the practice of dedicating certain young folk as "God's Watchdogs" or "Dogs" for short. These people travel from town to town, primarily acting as mail carriers and messengers, but with near limitless authority in matters of the church. The GM sets up the towns with some level of sin infesting it- ranging from simple envy to full blown murder and deals with demons. The players, taking on the role of the Dogs, have to ferret out the sin and fix it before it gets bad enough to destroy the town.
That sounds like it'd take a lot of the moral choice out of things- you are all basically playing wild west paladins- but the genius is in giving you so much power over matters of the church. Let me put this plain; If Bob is envious of Joe's girlfriend Sally, and you think the best way to fix this is shooting Joe in the head and conducting Bob and Sally's wedding yourself, you technically have the right to do that according to the minimalist doctrine provided. Sally or the other townsfolk might totally disagree with you, as might the non-religious authorities in town. (Say, the sheriff.) Honestly, the most common impediment to doing whatever the heck you want is the other players. Three Dogs working together to succeed at any cost is pretty close to unstoppable. The clever GM divides the protagonists, working to create a conflict between the Dogs.
The conflict system is where this shines. First off, there are four levels of conflict you can be operating at: Just Talking, Physical but not Fighting, Fighting without Guns and Gunfighting. Note that you can be operating at a different level than the other guy; I can be trying to talk you down while you're shooting at me. A mechanical bonus is offered for escalating, so if I can't talk you out of it then there's a strong temptation to force you to do what I want. We trade back and forth until we run out of dice or bow out; last man standing gets their way.
Damage is in the form of Fallout Dice. Assuming it doesn't kill you, Fallout can do several things, most commonly adding something to your sheet. Lost a knife fight with your brother? You might pick Fallout like a trait of "Angry at my Brother" that would give you a mechanical bonus when acting on it. The dice decide the severity of the results, but you decide the description, so "I forgive what my brother did to me" might also be fair. (GM does get to make sure it's reasonable. I've never had more than a few minutes debate over what Fallout is reasonable, and it doesn't get assigned until the fight's over.)
DitV is pretty hackable. By default, it runs from light fantasy (demons are real but subtle, prayer works sometimes but always invisibly) to high fantasy (Demons possess heretics and grow lots of nasty tentacles, shouted prayers become visible light that stops bullets.) Most of my time was spent in a Star Wars hack of it; all we did was change the setting and the articles of faith, and play jedi. No mechanical tweaking needed. (Okay, the Dogs all have to have a coat of office, and we used lightsabers instead. Does that count as a mechanical tweak?) I've also seen it used for Roman centurions in fantastic Britannica, and hit men working for the mob in a very Harry Dresden-esque world, so setting isn't really much of a barrier.
In regards to your edited in criteria;
1. Objective Standard of Morality, check.
There are rules of the Faith. Vanilla DitV lets the players make up quotes from the holy book that their characters retroactively always knew, but if you have an objective standard just make sure the players know it. (Or not. It wouldn't actually be against anything for a Dog not to have read the Book of Life, but just have been chosen by the elders anyway. Aaaaand now I want to play that character.)
Regardless, there is an objective evil force trying to corrupt the world, and every town you go into detail in will be in the process of falling into corruption. There's even a nice neat slider from "Mildly acting in discord with the scripture" to "Murdering the sheriff and consigning his soul to Baal" you can play with. If the players mess up and everything gets worse, just advance the slider to where ever makes sense.
2. Non-dualistic, check.
Last game of DitV I played, the players encountered a town that had a problem with Pride. The preacher's son thought he was ready to take up the robes, and his father was just holding him back so nobody would know how much better the kid was than the old man. The players screwed up hard, the dice hated them, one thing lead to another, and at the end of the session the church had burned down and the surviving townsfolk were being used as auto-cannibalistic hand puppets by the demon possessed youngest daughter via demonic shadowmagics.
That might have been unusually bad, but the best outcome of the demons winning is a lone serial killer murdering their way through the population until everyone dies of drought and disease brought about due to God abandoning them. No, I'm not joking. Assuming your players do not intervene, that is the eventual destiny of any town where even a single soul is taken in by the sin of pride. It may take a while, but it's going to happen. Good thing you guys are here to help, right?
3. Corruption worsens a character and virtue strengthens them, half right
There is no mechanical morality meter, just traits that change or get added or lost. These traits give a summary of the mechanical (and narritively powerful) characteristics of your character. Theoretically, no rule stops you from using every trait, no matter how weird, in every conflict. Likewise, you can take whatever makes sense as new traits, without a whole lot of rules oversight. Practically, you're limited to what the GM will buy, so acts of corruption should get corruptive sounding traits which can only be used when doing corruptivish things, and likewise for virtuous things. This means that your characters will usually get better and better at whatever they consistently do, until changing course is going to hurt a lot.
The Jedi in the Vineyard game, one of the party had a trait that started as "The ends sometimes justify the means 1d4." Then it was 2d4. Then 2d6, 3d6, 4d6, 4d8. He added a trait of "we are the judges 1d6" and shortly boosted it to 2d6. Nobody was really keeping track, you know? And man, he was dang good with his lightsaber. Best duelist in the party. Then, the inevitable; We needed a senator to change her vote, the woman's daughter was unattended at a party. "Sweet talk her, maybe ask her to talk to her mother for us!" someone suggested. A lousy roll- the man really didn't have much Acuity, and his Heart was only so-so. (The social stats.) He looks at the dice and goes "Hey, if I escalate and just kidnap the girl, that's Fighting, right? I'm better at that." The whole conflict, his traits kept pushing him to be more dangerous, to reach for his weapons, to justify his actions. In short, he fell, and hard , precisely because he was more mechanically stronger the closer he edged to the dark. By the time of the showdown, the only trait we had that could keep up with him was "My life for the good of the republic 5d12."
In short, DitV mechanically encourages you to be more of what you are. You can change course , but turning from dark to light will mean you get slowly weaker as you reduce your dangerous traits and build more rightious ones. It fails to make vice mechanically punishing, but I think it isn't an entirely meritless failure.
4. Personality, check.
Is a trait of "The ends justify the means" an evil thing? The fallen Jedi certianly misused his, but that phrase is at the core of utilitarianism, which is generally a pretty "good" philosophy. At the very least, there can be as many different takes on Good in your game as your objective faith allows. Or there can be characters who aren't really attached to Good and Evil- maybe all your traits have to do with how much you love roving the wilderness and how awesome you think your master is. My jedi's biggest trait was "Shoes are prisons for your feet." A moral philosopher I was not.
5 Traditional RPG, probably check.
I feel really weird calling DitV a traditional RPG. It's not even remotely balanced, talking things out comprises a whole quarter of conflict and actual murderization is only half, it is leaning towards minimalism (the whole rulebook, including fluff, is 150 pages, and you could probably fit the conflict rules on two sides of paper) and it is crazily narrativist, enough to send my simulationist tendencies gibbering into the corner.
Nevertheless, it has fighting and magic as a core part of the rules, it is often about going into a new place and punching things until they stop being bad, there's one GM and a bunch of players who each control a single character, etc. I don't know exactly what constitutes "mainstream elements" but if FATE counts, DitV probably also counts. The main thing I would warn you about is the high likelyhood of PvP, which may be unusual in traditional games. (I wouldn't know, having died by 'friendly' fire more often than I've died by GM powered dangers in my traditional play.)
Final score, 3 and two halves stars out of five.