On the whole I agree with you, christinarossetti about not getting a present for the child you don't know, or only a token one, but I'm only writing from my own experience and circumstances of my dc's class.
All the kids in the class know each other, and it is with another child from this class that the joint party is with.
So all the invitees know the birthday kids very well.
In this situation, the invitations said " x & y would like to invite you...." Etc.
Therefore, in this situation I think that a present should be bought for each child, as both children are inviting the others.
In the case of the op, she should probably clarify with the parent of the underlined child if she was specifically invited by that child.
If both children invited, then in etiquette terms, both should have presents. If only one child invited, then the present should be bought for that child.
I also don't understand your comment about "that's their lookout" if parents choose to split the cost of an expensive party. Lookout for what? I'm not saying that the kids should get presents because it's an expensive party.