Here are three alternative ways to rephrase "Your request is denied," aiming for enhanced empathy and clarity:

I understand how much this request means to you, and I am genuinely sorry to say that after careful consideration, we are unable to approve it at this time. We appreciate you bringing it forward.

Thank you for sharing your request with us. After a thorough review, it is with regret that I must inform you we cannot fulfill it as presented. We truly value your input.

We have given your request thoughtful consideration and understand your perspective. Unfortunately, we are not in a position to grant it at this moment. We appreciate your understanding in this matter.