These words all begin with 花（はな↓）, but they all have different pitch accent patterns.
花火 は↓なび (There is also 花火↓師)
花見 はなみ↓ (also, 花見↓月 vs 花見時 which is 平板)
Knowing that the pitch accent of each word is 火↓、芽↓、見↓る（also, 見↓事) does not help me identify a pattern either. What are the rules that causes the words to change their pitch accent in compound words?
All references for pitch accent come from スーパー大辞林