The first clue to choosing the correct expression to go in the blank is to note that "that mother" is the topic of the sentence because it is marked with は
So, unless another actor/subject is explicitly expressed, the mother's actions and thoughts are being expressed here.
The next clue is the verb 困る. Something is potentially going to annoy this mother.
Given these two clues, choices A and B make no sense in that position. Whatever verb you fill in here needs to retain "mother" as the subject. Choices A and B would seem to require 私 as the subject. It is not natural to switch the subject of the verb like that in the middle of a sentence.
But this is particularly true given the next clue
You questioned the meaning of そう. This cannot be the hearsay そう because the grammar is incorrect for that form. If you wanted the hearsay そう then the grammar would need to be
But that really makes absolutely no sense
A face that I hear wants to say she'll be upset if ...
What's being described is a scene in which the mother is lecturing me. I don't require "hearsay" to see the expression on her face.
So, the only two choices that make any potential sense are choices C and D.
C takes the wrong perspective.
So, the only answer that works is D. Keep in mind that くれる takes the perspective of the subject being quoted. Here one is quoting the mother's perspective--what the mother is "saying" by her facial expression. Only in choice D is it stated clearly that if you don't heed this mother's admonishment she'll be upset.