Has it been confirmed that the auto-save feature is enough to overwrite the ilvl in-game?
I guess I was unclear as usual.

Sorry!
...but that would not affect the item in-game in real-time, and therefore wouldn't go in affect until you leave the game.
No, the auto save does
not affect the item in-game. Saving just updates your character file on disk, but it changes nothing about the game you are in.
But what happens when you use ATMA or GoMule to hotmule the item, I wouldn't know, as I don't have/use them.
Interested to learn if time is the only trigger for the auto-save character function.
As far as I could tell the only other triggers are quest-related. But I didn't look for long, so I could easily have missed something.

But the auto-save won't overwrite the ilvl of the item
in-game, so if you use the rune in the game it was found in, it'll be good.

No need to hurry, and no need to keep any save-triggers in mind.
Edit: Maybe good to mention that the ilvl of crafts gets capped to 99, so if your rune came from LK, the ilvl will already cap out when you're clvl 63, and there would be no benefit to being higher level (sorry pharphis!). I didn't mention it before because I thought that ilvl 65 was the maximum anyway.

Oh the good ol' pre-99 days, eh?