"내가 왜 거짓말을 해야 돼요?" is a fine sentence but it sounds too serious here. It sounds like one is forced to tell a lie for whatever reason. 거짓말 seems too serious for the occasion. What you described would probably go like this (assuming between close friends) in real life: A: 너 오늘 진짜 멋져 보인다. (You look really awesome today) B: 정말? 너 지금 나 놀리는 거지? (Really? You are teasing me, aren't you?) * 놀리다: tease; pull someone's leg. A: 아니. 내가 왜 널 놀리겠어? (Not at all. Why would I tease you?)

"해야 돼"라는 의미보다는 "해?", "하겠어?"의 의미 아닐까요? "너 지금 나한테 거짓말 하는 거지?"( You are telling a lie to me? I do not believe what you said to me." "내가 왜 너에게 거짓말 '해/하냐/하겠어/하겠냐'?"(I am not telling a lie to you. Believe what I have said now. I am not a liar.)="나 거짓말하고 있는 것 아냐!", "내가 한 말 거짓말 아니야!" In the above sentence, 'should' does not have the same meaning as 'aught to' or 'must'. -How should I know where you've left your bag? 네가 네 가방을 어디에다 뒀는지 '내가 어떻게 알아?' -Why should he say that to you? 왜 그가 당신에게 그런 말을 하는 것일까요?

In word-to-word translation, Yes. But what is the situation(contextually)? Does someone want you to lie? Then, Yes. Do you want to express your thought (I don't want to lie.) by "Why should I lie?" Then, Yes.