@davidhohn this is an incredibly interesting thread! I find myself asking things like this a lot because me and my partner write and illustrate as a partnership (they are mixed race Scottish/Indian - from India not Native American and I'm boring old whiter than white). I illustrate their work where race is not a focus of the story. ie, they are writing a book of fantastical short stories that never brings race into the narrative directly - therefore, I'm illustrating it. However, they are also writing a book specifically about being mixed race - highlighting some of the awesome mixed race people in Britain - and I deliberately said "I think you need an illustrator who is of colour for this one" because I personally feel that it's not my story to tell.
In the case that you are talking about, I think you absolutely must approach a project like that with caution - especially if you are writing and illustrating it with no partnership/input from a person of the culture you wish to depct. I agree that you can be curious and extremely knowledgeable about a culture that is not your own but you must recognise that it is not your own and no matter how much you research you're still not going to know as much as a "native" person might. A good example is "William Dalrymple's City of Djinns." He is white but writing about Indian culture and though he is well revered, he still comes across as a bit colonialist - in my opinion. You could even blatantly mention in the book that you are not of that culture (I personally like this approach if you use the right tone, however, I can see this having problems too).
Another thing I came across recently is "Diversity Readers". This is when you hire someone of that culture/group (LGBTQ+ groups are included in this) to read the work you have created and give opinions as to its suitability from their perspective. [DISCLAIMER; in my opinion, this should be the last step in any person's thought process when trying to reckon if they should write/illustrate about someone else's culture/group and not a way to hurdle these issues and simply state "well if I get someone of that culture/group to read it I've done my best and now it's ok for me to write about them." because this could just turn into another way to quiet voices of people from underrepresented groups.] I think another thing you have to be aware of is that you may well do your darn best to represent that culture and be inclusive and even have it checked by someone of that culture and still have a negative reaction from the group you are portraying. This is absolutely their right! You are, when all is said and done, still writing about a culture that is not your own and if some members of that culture don't like it, you have to remember that you are the one who decided to tread there.
In conclusion, it's a very delicate balance between appropriation and appreciation. Sometimes just because you appreciate something doesn't mean it needs to be your voice telling that story. I think we should all be asking ourselves "is this my story to tell" a lot more.