Your gouge seems to avoid the main contact areas where the screws will go, so is reasonably cosmetic. Grain filler doesn't have much structural integrity in large amounts, so I think a normal wood filler would be best.
In terms of looks (as you should try to avoid painting in the neck pocket except around the very edges), you could consider filling the gouge, then sticking a piece of ash or maple veneer over the top. It will raise the neck up slightly, but not by much (and you can always sand the veneer even thinner first).