Large language models (LLMs) enabled conversational robots to shift toward free-form interaction. However, without context-specific adaptation, generic LLM outputs can be ineffective or inappropriate. This adaptation is often attempted through prompt engineering, which is non-intuitive and tedious. Moreover, predominant design practice in HRI relies on impression-based, trial-and-error refinement without structured methods or tools, making the process inefficient and inconsistent. To address this, we present AI-Aided Conversation Engine (ACE) to support deliberate design of human-robot conversations with three key innovations: 1) an LLM-powered voice agent that scaffolds initial prompt creation to overcome the "blank page problem," 2) an annotation interface that enables the collection of granular and grounded feedback on conversational transcripts, and 3) using LLMs to translate user feedback into prompt refinements. We evaluated ACE through two user studies, examining both designs' experience and end users' interactions with robots designed using ACE. Results show that ACE facilitates the creation of robot behavior prompts with greater clarity and specificity, and that the prompts generated with ACE lead to higher-quality human-robot conversational interactions.