I'm a native speaker. This is my speculation, but I believe more or less it's the truth.
First, "business" was a little difficult (as it had three syllables) for most Chinese people at that time, so they would like to miss the third syllable, only pronounced it like "busin".
Second, both Mandarin and Cantonese (no matter now and then) did not have the exact /z/ sound, so we would transliterate it into the /tɕ/ sound (aka
j in pinyin) – similar situation for Japanese, they say
ビ(bi)ジ(ji)ネ(ne)ス(su) for "business" – Unfortunately, the /dʒ/ sound followed the same way because there's no /dʒ/ in Mandarin and Cantonese either. So, even today, when I hear "the letter /tɕiː/" from a Chinese native speaker, I have to ask whether it's a G, Z, or even J.
Finally, the Chinese people at that time would pronounce "business" like /ˈbɪtɕɪn/, and then Europeans could not distinguish it from "pigeon" with a similar Chinese accent (the reason for /b/ to /p/ might also be the influence of the "Wade–Giles system", which transliterated the
b sound in pinyin back to "p". Peking vs. Beijing is a perfect example): as a consequence, "pigeon" and later "pidgin" was used for that pidgin word.