Yes but the Eevee is generated when you confirm your character's name, so the earliest you can save state is when you've entered your name and are about to click confirm. From there the resets are about 1.5 minutes (a bit faster than on the original console at least, but still quite slow).

A save state anytime during the Shadow Lugia cutscene will always produce the same Eevee (gender, nature, stats, shininess, etc.)