Why is it 만나서 반갑습니다 and not 만나니 반갑습니다? The verb is clearly in 하십시오체, so shouldn't the 'Cause/Reason Connective Form' (as wiktionary calls it) 서 also be of the form 니?
It is not true that -(으)니 is associated exclusively with 하십시오체, nor is -(아/어)서 associated exclusively with 해체. In fact, these causal connective forms are used pretty much across all speech levels, although there is a distinct poetic / old vibe to -(으)니.
This blog post gives a good summary of the uses for -(으)니. You can see that it is more related to -(으)니까 in its causal meaning. The distinction between -(아/어)서 and -(으)니까 I'm sure you're familiar with, as it is a fundamental piece of Korean grammar.
One of the links above also states how only -(아/어)서 is used in "basic greetings". I perceive that because -(아/어)서 is for "naturally" arising cause-to-effect relationships, only that form is appropriate for 만나서 반갑습니다.
In addition to Michaelyus's nice answer, one more thing to consider is that 만나서 반갑습니다 is a fixed phrase.
Consider English "Good morning": you don't say "nice morning" even though good and nice are pretty much interchangeable, because it's a fixed phrase. Same for 만나서 반갑습니다.
- Also, sentence-connecting(?) suffixes like -(으)니 aren't really associated with speech levels. You can have a perfectly natural expression with -으니 at any speech level, e.g.,
해가 지니 추워/춥네/춥다/추워요/춥습니다/etc. = It's cold now that the sun went down.
그렇게 하니 망가지지(요)! = You are doing it like that, that's why it breaks!