You've said yourself - he isn't Aleut. By definition, he's a cultural outsider. One who went to primary sources for research, but still an outsider. That doesn't mean the book has no use, but you shouldn't use it as a single source. (Or any other book for that matter... you need as many sources as you can find, and beware sources repeating inaccurate information from other sources).
As well as what happens in the story, you also need to know what it means to the people who tell it. The story may vary between different communities. It may be one that's only told at a certain time of year or not one that's normally discussed outside. You may be better off hinting at aspects of the story, rather than directly retelling it.
So read that book, but don't just read that book. Look for critical essays and reviews about that book, especially those by Aleut people. Find other books by outsiders about Aleut people, and see what criticisms were raised about those. Read books that are written by Aleut people (actually by them, not through a third party), and again, look for criticism (just because it's written by an insider doesn't mean everyone will agree it got things right).
Read things that you don't think are directly connected, rather than just folklore books. Blogs by Aleut people on their daily life, news stories, forums. Read any fiction and poetry by and about Aleut people you can find, regardless of subject. When you don't know about a culture, don't assume you know what it is you need to research. Read widely.
And don't assume you know if a source is accurate or not. You're defending books here, when you've said yourself you don't have that knowledge. Question what they're telling you. Be prepared to discount a source if it becomes clear it's shakey.
You'll still make mistakes, but far fewer.
