The clause preceding と, i.e., quotation is the addressee's words, so it's like これを食べろというのか and あそこに行けというのか in the thread linked below. It's just not the quotation is not imperative, i.e., not an order/a request to the speaker here.
「で、ではどうしろというのだ！もう臨界状態なのだぞ！」 Hi. I know this というのだ is an exclamation marker. But can I think this というのだ literally means “you say what to do” or “you tell me what to do”? Thank you.
As Chris-san already wrote, it's a rhetorical question, not a real question, so it means "Are you saying ~? (= "I can't believe it. It must be wrong." or "It's unbelievable, but it could be true.")"
As I wrote in the thread linked above, this というのか can't be omitted (at least と/って is necessary) when the quotation contains an interrogative like いつ, 何, どこ or どう. However, if these words are not in the clause, というのか can be omitted by saying it in a rising intonation.
その原因が、あの少女？
その力を人間の身に封じて、何も異常が起こらない？
これを食べろ？
あそこに行け？
(It's more common to put と or って at the end of the sentence.)
×どうしろ？
×何を食べろ？
×どこに行け？
(At least と or って is necessary.)