Izdubar at Large

Question: If Gilgamesh had reached true apotheosis, what kind of stats would he have?

Impossible to say. Gilgamesh wasn't even close to apotheosis when he died - I'd peg him at Legend 5-7 or thereabouts, which means that trying to figure out what he might have been god of is pretty fruitless. He could have developed in any number of directions or had any number of changes of heart or adventures that displayed or encouraged new skills; believe you me, I have never seen a Scion in play at Legend 9 that could have been accurately guessed from their stats at Legend 6. What would Gilgamesh have been god of? What would people have worshiped him for? What other things would he have done to achieve godhood, and what would they have involved? We don't know (and can't, because he died), so attempting to guess what he might have grown into is just that: guessing.

If I were trying to stat Gilgamesh as he was at the time of his death, I'd probably give him large quantities of Epic Strength (for his feats of building, fighting and wrestling Enkidu), Prophecy (for the many prophetic dreams he has about friends and enemies he's about to meet), Mystery (for the universal secrets he gained from Utnapishtim), and probably Epic Charisma (because no matter how enormously dickishly he behaves, people seem to love this dude anyway). He is also set up after his death as a judge of the dead in the Underworld, so you could give him some Justice as well (though watch out - he has quite a reputation for tyranny).

As far as whether he would have carried those things on into godhood, though, we'll never know, and you might as well make a guess at whatever you think would have made the most sense. If you're planning to use Gilgamesh as a deity in your game, consider coming up with some other things he did in the interim that caused him to become a god, so that it's not a surprise jump from Demigodhood to full-blown Scion-fathering craziness. You might find that some of your stories for his exploits lead to other skills he's more suited to have as stats (for example, if he escapes Irkallu through sheer determination and moxy, you might determine he has some Psychopomp). But keep in mind that this would all be invented stuff, not mythically based, so the flavor of your game might be affected for those who are up on their Babylonian epics.