You are correct. We must not allow ourselves to be exploited through that.
First, we need to try to reason with the client. But not in a chatty way. If they blackmail you, get up from that chair, cool down. Then, return when you are calm and respond with a single well-worded message.
If that fails, involve customer support.
And if that fails as well, we offer the cancellation.
And let me underline something. Cancellations have a private survey too. I am not sure how it impacts us. I am a seller like you.
@carrieblogger has some good videos about customer service and cancellations on YouTube.