A couple ideas to consider:
If you labeled your icons here (generally good practice but not always an option), would the menu icons have different labels? If not, I'd be very careful about using two symbols to represent the same thing. You're establishing a language here, and you don't want people to be confused about which words you're using.
To me, kebab menus (...) symbolize optionality or "more" (kind of how an ellipses is used in writing) compared to hamburger menus, which generally symbolize global navigation. If you're going to use a single icon, the kebab seems like the more appropriate choice in the context you described.
One reason for the awkwardness you mentioned is that the child items feel heavier than the parent because the icons are filled. They naturally draw the eye more than the parent icon. Do they directly contribute to the usability of this menu? If not, I would trim them (I'd say the same for all icons, actually).
Don't be afraid to use opacity to increase differentiation between elements. In this case, you may be able to lighten your menu icons to help things feel a bit "lighter." Be sure your contrast is high enough to be accessible.
Taking all this ☝️ into account: