There is an inherent contradiction in any Paladin fighting evil wherever it is, as a basic world view, and consistently performing as an Evil player character. It will or would take considerable gymnastics by both the GM and the player to establish an environment wherein the oath keeper fights his own nature.
In the case of the Lawful Neutral confronted with supporting the rules of a Lawful Evil society, what's the rest of the story? What's the context? What's the greater threat to both societies? (Consider alliances in history: in the 1930's, Red Baiting and Anti Communism was rampant in America. In the following decade, an alliance was made with Communist USSR, a totalitarian society, to fight another totalitarian society. That didn't make America oath breakers, nor totalitarian).
Further example: if your Lawful Neutral Paladin is human, and the Lawful Evil society is as well, and the threat to both is from Hobgoblin hordes from the Western mountains, what's the greater threat? What's the common bond?
The loophole to all of this is the "greater evil" theme.
For a campaign it amounts to a limitation on the viability of the character: once the greater evil is defeated, he has to confront the lesser evils, which will eventually include the evil within himself. (See Elric of Melnibone as a modest example).
Paladins fight evil, wherever they find it. If it is resident within the Paladin, he has to take on that struggle.
This appears to have a game play limitation that has to be resolved by ... alignment change, oath breaking, or hanging up the cleats and retiring from the profession.
Suicide is also a valid resolution to this internal conflict, but I doubt that makes for satisfying role playing, nor is that likely the direction the game's authors intended to go with Paladins.
I will ask this question of those trying to create a Paladin with evil alignment:
Why are you going out of your way to create a self-contradictory character? Why are you making this hard on you and on the GM? Think of the other players, perhaps.
The new rules seem to be an effort to get away from the old "lawful good equals stupid" paradigm, and also tries to make the Paladin a better role for a mixed class/mixed motive/mixed alignment party without becoming an oath breaker over one of a hundred little details that come up when parties go on quests or adventures.
Focus on The Party or The Group: the D & D model is built upon a band of adventurers, each with something different to offer, putting their efforts together (hopefully achieving synergy) to overcome a series of challenges.
As a Paladin might tell you: it's not all about you.
It's about fitting into, and possibly leading, the party or the group. Remember our basics about Paladins from the very beginning: high charisma is a requirement. That is force of personality, leadership, etc.
Back before 1st edition AD & D, when the Paladin was introduced in the Greyhawk supplement, and in 1st Edition, a core problem with role playing a paladin is that a Lawful Good Paladin was stated to not generally hang out with anyone of evil or chaotic alignments.
It is good to see that the framework in 5e has overcome that Game Play (meta) problem.