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)
花芽 はな↓め
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
This is unfortunately exceptionally complex and irregular, but there are essentially patterns which you can carve out of other patterns and then out of those carve more exceptions.
The first pattern is that a two-mora + one-mora word generally becomes atamadaka (maybe 60% of words?). However you need to carve out a huge number of exceptions. But that is the default.
花火: So, は＼なび falls into that pattern with no need to think of it being exceptional. It avoids the kunyomi-compound pattern I mention later due to being rendaku’d.
花芽: Very often, N+N kunyomi compounds (particularly when they are not rendakued) can result in a drop between the two kunyomi words — this gives the feeling that the two words are still somewhat distinct, not completely merged. I’d say this is what はな＼め falls into. (Compare やまがわ￣ (a mountain river) vs やま＼かわ (mountains and rivers)). This is a very common pattern, like つき＼ひ, はる＼かぜ, etc. Perhaps even worth thinking of it as the default.
花見: はなみ＼, つきみ＼, ゆきみ＼ are sort of their own class of class of exception. In general N+V kunyomi words become heiban, but you occasionally see some odaka words when V is one mora like 足蹴＼ 雨着＼ 春着＼ , but even the one-mora V words tend to be heiban so these are still exceptions.
花火師: 〜＼師 , being the one actual suffix you listed (as opposed to just being the second half of a word) has far more consistent pitch: it always causes the word to drop before the suffix （看護師、伝道師、道化師 etc). The NHK accent dictionary lists verb many such suffixes on their own. Some cause a drop before the suffix, some cause the word to become heiban (〜的、〜上（じょう）、〜語, etc), some cause it to become odaka (〜後), some a drop in the middle of the word (like 〜人 in its secondary pattern: にほんじ＼ん, ちょうせんじ＼ん, たいわんじ＼ん, which seems to activate when the word it connects to is odaka. The normal pattern is ＼じん though: アメリカ＼じん、ちゅうごく＼じん、かんこく＼じん)
In the end there are so many exceptions that the only way to succeed is to train your ear, get a ton of exposure, and get corrections; but noticing patterns and storing exceptions more efficiently can be a decent stopgap towards building a useful feedback/correction loop for yourself along the way. Eventually these just turn into “intuition” (probably).
