The Cantonese for "Canada" is gaa1 naa4 daai6 in Jyutping transcription (your transcription of "Mexico" is fine though).
It feels safe to say the second syllable beginning with N in Mandarin was just good luck though. I might have swapped out the N for L anyway if I had known the Jyutping.
An example of a distinction that would have come through fine English -> Mandarin being lost by apparent transmission through Cantonese exists in Los Angeles, 洛杉矶. The Mandarin is luo-shan-ji, which annoys me every time I need to understand or produce the name. Mandarin has no problem distinguishing s from sh.