The first syllable 'Ca-' rhymes with 'hi', 'bye', 'I' etc.
The second syllable '-mi' rhymes with 'me', 'bee', 'see' etc.
Is this correct?
If so, I think your interpretation, Kayme, is good. Or perhaps Kaime, as some people may pronounce Kayme as 'came'. Even just adding these into the original might be good, and doesn't change it too much - Caymi, or Caimi. The use of '-e' at the end of a word (as in Kayme) is, as you know, often silent, and functions as an indication of a change in vowel sound (cam - came, Sam - same). Whereas the '-i' at the end (as in Caymi) would indicate that it is to be pronounced to someone who read your name. Just a small thing.
It's an interesting name, Cami, and I can see the frustration you'd have if it is pronounced so differently from how it looks.