I've found in this site
that ~あて can be used with meaning "per" (jisho confirms it as well), and the site shows this example
一人【ひとり】あて三冊【さんさつ】 = three books per person
However, most of Japanese natives I've asked say this sounds unnatural (but cannot explain the reason, but mostly by "internal sense of naturality") and that they would say it with ~あたり or with ~につき.
Then I would really like to know if that example is indeed unnatural or not, and if so, the reason why.
Besides, I would like to know in which cases I can use ~あて interchangeably with ~あたり and ~につき and to see some examples if possible.