The Problem with Paladin Villains (spoilers for SoD/GDGU)

[Worldsong]

No, no, and no. A lawful good deity understands that devotion cannot be compelled. Therefore a lawful good deity will not mandate worship, and certainly won't allow his or her followers to hand out draconian punishments for those who won't be coerced. You are describing a lawful neutral or lawful evil system that wants to call itself lawful good. Any paladin working for such a system would fall.

A good DM could still put together the scenario you describe, with the empire claiming that they do have real paladins, but there would be no true paladins on the side of this empire. The clerics would be deriving their spells from some other god, possibly even while being deluded that they were worshiping the lawful good deity.

I think you might be making some assumptions about what are mandatory requirements for a deity or person to be Lawful Good.

Actually that's not entirely fair. Your interpretation of Lawful Good is valid but I don't think it's the only one and if a DM made such an empire and said it still qualifies as Lawful Good I'd see it as an example of how Lawful Good can end up being problematic rather than it being not Lawful Good.