The Oathbreaker paladin subclass from the DMG is designed for evil characters, but you have other options
As you have noticed, the flavour of the class features and oath spells for the Oathbreaker (DMG, p. 97) are heavily leaning towards an evil character*, which makes sense given that it's under the Villainous Class Options section.
However, it sounds as though there might be another way to match this backstory with the paladin class.
You mention that:
He is actually talking about writing his own treaties down which will be the opposite of everything his god believes.
This, to me, sounds a lot like an oath, which is what paladins are all about. Ok sure, he's broken whatever oath he may had sworn to this old god, but he seems to have sworn (or be interested in swaring) a new oath. This makes him a regular paladin, not necessarily an oathbreaker (even though he has also broken his previous oath).
Also, remember that in 5e, a paladin's powers come from their belief in their oath, not necessarily a god (paladins in 5e aren't tied to gods as much as in previous editions):
Although many paladins are devoted to gods of good, a paladin’s power comes as much from a commitment to justice itself as it does from a god.
Personally, given other clues in your question, such as:
we have said that the player will be neutral when the campaign starts and still having to consciously not slip back, there is a clear arc here where he can then find his own redemption and become good.
So the Oath of Redemption subclass sounds like it might be a good fit.
But really, any of the subclasses (that aren't Oathbreaker) would fit, depending on the nature of the paladin's new oath, that detail is up to the player (and you, the DM).
Don't feel like breaking an oath necessarily forces you to use the Oathbreaker subclass. You are in control of the rules, not the other way around.
* I won't say that you have to be evil to use the Oathbreaker subclass, because, although it leans towards evil, someone can make a character for whom those evil-ish abilities make sense without them having to actually be evil, if they can make it work. But the implication is that Oathbreaker paladins are probably evil, given the description of them in the DMG.
Best Answer
Figuring out the answer to your question has to start with examining the intent of the authors of D&D 5e. When we look at the paladin one of the first statements we find is this.
The author then go on to explain that paladin uphold a cause of righteousness.
These are both in the initial flavor text which applies to all the listed variants of paladins.
The different Paladin Oaths are focused on motivation. Why does the paladin act as he does while pursuing a cause of righteousness.
Does he love the ideals of justice, virtue, and order? Then his motivation is likely a Oath of Devotion. Do they love life and the beauty of living things, then he is best taking the oath of the ancients. Does the paladin view himself as the instrument of punishment against evildoers, then the Oath of Vengeance is suited for the character.
D&D 5e has the ninefold alignment system. Good and Evil are clearly different things especially in terms of the cosmology. A Lawful Neutral Paladin may view order and rules vastly more important than justice but as a Paladin his primary focus is still the Cause of Righteousness as stated in the description of the class. Where the Lawful Neutral Palaadin differs from the Lawful Good, Neutral Good, Chaotic Good, and even Chaotic Neutral paladin is in what should be emphasized and focused on while pursuing the various missions in support of Righteousness. But none of this will lead to a Lawful Neutral paladin to support the cause of Evil even when it is Lawful Evil.
To recap, the paladin's oath speaks to the paladin's motivation. The paladin's alignment speaks to paladin's focus and means.
What is subjective is the referee's definition of good and evil for his campaign. Individual referees will set the lines between good and evil in different ways. But once it set for the campaign then those are the lines that a even a Lawful Neutral paladin will not cross lest he becomes an oathbreaker paladin.
Evil Paladins
In creating a paladin we get this statement.
So how could a paladin that is any of the three evil alignment be pursuing the Cause of Righteousness.
Remember earlier I said that for Paladins, alignment speaks to focus and means. A Lawful Evil, Neutral Evil, and Chaotic Evil paladins will likely have an attitude of the ends justify the means. In their minds they separate the world into innocents and evildoers and anything goes against the evildoers including methods that paladins would find abhorrent. The line for these evil paladins would be whether their actions effect innocents. Anybody straying across that line is in their mind fair game.
A Lawful Evil paladin would not uphold the laws of a Lawful Evil society that promote tyranny and injustice. Instead they would view that order and discipline are vital tools in combating those who do evil, however the campaign defines it. They themselves would resort to torture, lies, enslaving and manipulation of evildoers to achieve their destruction.
To use examples from comics book think of the differences between Marvel's Punisher, DC's Batman, and Superman.
I will end by saying that it is definitely a challenge to roleplay a evil paladin but still uphold the cause of righteousness. But the RAW rules do not but a restriction on it.