Prior clinical psychology research shows that Western individuals with depression tend to report psychological symptoms, while Eastern individuals report somatic ones. We test whether Large Language Models (LLMs), which are increasingly used in mental health, reproduce these cultural patterns by prompting them with Western or Eastern personas. Results show that LLMs largely fail to replicate the patterns when prompted in English, though prompting in major Eastern languages (i.e., Chinese, Japanese, and Hindi) improves alignment in several configurations. Our analysis pinpoints two key reasons for this failure: the models' low sensitivity to cultural personas and a strong, culturally invariant symptom hierarchy that overrides cultural cues. These findings reveal that while prompt language is important, current general-purpose LLMs lack the robust, culture-aware capabilities essential for safe and effective mental health applications.