My actual objection to that argument is this: If men and women are essentially different, why do we assume some sort of indoctrination is necessary to either reinforce that or to make them so? Will they not turn out different?

Regarding your prior comment your definition is what I’m comfortable with. I think the real argument here is not to discover what men and women are like, but rather to understand what relevant scriptures have to say to us.

1. If we are not complementarians, we will not train men to be men and women to be women.
2. If we do not train men to be men and women to be women, then there is no way according to which we are training people at all.
3. Therefore, if we are not complementarians, there is no way according to which we are training anyone.

There are plenty of things complementarians might accuse egalitarians of doing, but it’s pretty crazy to think they follow nothing in the Bible and do not train their children in any righteousness at all simply because they don’t accept complementarian gender-related stuff. You say he’s got an irenic tone that promotes unity, but he’s smuggling in a pretty serious accusation against egalitarians with this argument.

In the other direction, I can imagine an egalitarian who thinks women should be prominent in preaching and in authority over men in the church and that men and women should be equal in decision-making and authority in marriage but who thinks all manner of other characteristics are gender-determined in a way that would still count as essentialist.

So I think essentialism is neither required for complementarianism (as Carson and Baugh demonstrate) nor incompatible with egalitarianism (as your case shows, and my suggestion makes even more clear).

A better way to define complementarianism is that it’s the view that God has specifically limited women’s roles in ministry and marriage, usually in the case of the church in terms of authority over men and/or teaching over men, and in the case of marriage with some kind of authority or submission that isn’t fully symmetrical. How and why God does this will vary from view to view, although I’d say it’s no longer complementarianism if it involves a view of ontological superiority of men.

Egalitarianism, I would say, is the view that God has not limited women’s roles in the church or home in a way that makes them asymmetrical to men’s roles, at least in theory. (Egalitarianism is compatible with any particular congregation having a man in authority over every member.)