After reading pretty much every post on というの I can find, I'm finally giving in and making a new one. Hopefully making this will help others who are searching for it.
I am aware of using というの to deifne things (犬というのは何？もじゃもじゃしたトカゲ？）
I am aware of using というの to give a reason / comment on a situation（宇宙人が来てるというのは、電波機が働いてるってこと？）
I understand that at it's core it's just a quotation device plus a normalizer.
What I want to understand is the reactive というの that's used often at the end of sentences. The most simple example I can think of is "何と言うの！？"
The part that's confusing me is how much of this という is quotative, and how much is summarzing or commenting on facts.
Here's my example sentence. This occurs as two characters are fighting (character B), and a third character (character A) watches on.
A：「妾はどちらを応援すれば良いというのか。」
B：「こいつっ、私と互角に渡り合うと言うの…！」
My confusion is the ambiguity between という as summarzing a fact/idea and という as a quotation device. I feel like I could read this as "You're saying to fight evenly-matched?! (yeah that's a bit awkward in English but w/e)… or as "It is the case that we'll fight evenly-matched!?) and I don't know how to parse the difference.
Is it simply that there actually is ambiguity there and that 言う instead of いう is just a hint that it's quotative? If it is a quotation / character B is questioning what character A said directly, why is it then not と言ってるの！ instead?
(And as if the author is trying to confuse specifically me, there's another というの in literally the line before, though that seems more clearly rhetorical.. maybe)
I realize I'm splitting hairs a bit, but given how fundemental this grammar is, I'd really like to understand it more deeply.
EDIT: Thanks for the responses everybody, this was extremely helpful.
by Enough_Tumbleweed739