Is it possible to identify click sounds like [‖ ʘ !] by formant transitions in the surrounding vowels? I know stops and fricatives have that feature. I'm just wondering how the five (main) click sounds differ from each other acoustically.
I tenatively propose that you can not, and that click identity depends very heavily on the burst. I don't know if anyone has run the experimenent, but you can sort of start with recordings of Nama, Xhosa or !Xóõ, edit out the burst, and play the result. Ideally you'd want minimal pairs (and actual speakers of the language for your subjects), but you can get a quick DIY answer from these materials. My opinion is that click place cannot be identified at all, not just limiting oneself to formant transitions, but some of the fancy features like epiglottalization might have an audible carry-over past the burst though not something realized in the pattern of formant transitions.
EDIT: I'll be less tenative. Tony Traill's 1997 paper Linguistic phonetic features for clicks ran the experiment with !Xóõ speakers, and found this result. He notes that the anterior constriction does not coarticulate with the following vowel and the posterior constriction coarticulates with the preceding only to the point of indicating some velar constriction.