If I knew you well, and were trying to recruit you, I'd probably let you do it. I wouldn't like it, but I'd do it.
In pretty much any other scenario, I would consider it a dealbreaker.
Allowing an interviewee to record the answers to your difficult, sensitive interview questions is a significant risk to the reputation, and possibly the legal standing, of a company. A malicious interviewee could do, at the least, serious reputational damage by taking our responses out of context.
As soon as the risk of damage becomes higher than the reward of possibly hiring you, it's not worthwhile to continue the interview.