Any “fai” (zy: ㄈㄞ) pronunciation record in Mandarin?
Not in standard Mandarin, no.

But there are regions in the South where Mandarin phonetics are not rendered standardly, and Standard hu- (also sometimes ku-) tends to be pronounced f-. In Western Fújiàn I often heard huàidàn 壞蛋 'bad person, asshole, etc.' pronounced as though the Pinyin were fàitàn. But I wouldn't count on using that to write "WiFi".

An example of regional f- for Mandarin k- appears in New York's Chinatown, where Forsyth Street is translated Kēxījiē 科西街, because the community was completely Cantonese-speaking at one time. You can see it on the street signs.