I'll answer with the aid of Osage, since that's the language I know (my father's people are Osage) and have good resources for. Osage is a Western Siouxan language, closely related to the Lakota spoken by the "Sioux" from your story1. So while this isn't the answer for them, it should at least give you the right idea.
The other big advantage to using Osage here is that we have a dictionary for it written by an actual Native Anthropologist, Francis La Flesche, around the turn of the 20th Century, not that far from the period you are asking about. He was born among the Omaha in the decade before your story. Omaha is also a Siouxan language, very closely related to the Osage language.
Since English wasn't their native language, one would imagine when asking such a thing in English, indigenous peoples would use the appropriate English words, which in this case would be "Indian".
In their own languages, they'd likely use their own endonym for themselves, often translating into something like "the people". For my father's people, it was 𐓏𐒰𐓓𐒰𐓓𐒷 (Wa-zha'-zhe, simplified by the French to "Osage"). I've seen it translated as "People of the middle waters", but La Flesche writing in the early 20th century said the exact meaning had been lost. For the people who Kelley called "Sioux", their endonym is "Lakota" (which I will hereafter use, as per their wishes).
As for exonyms, the Osage use 𐓐𐓎𐒹𐒰𐓆𐒼𐒰 (xin-ha' çka) for people of European descent. Its a compound word meaning "white skin"3. According to La Flesche, for (American) Indians in general the Osage word was and is 𐓁𐒻𐒼𐒰 𐓓𐓎𐓊𐒷 (Ne-ka zhu-dse). This is a compound word of "red man". However, La Flesche reported that they had as an antonym to 𐓐𐓎𐒹𐒰𐓆𐒼𐒰 the word 𐓐𐓎𐒹𐒰𐓆𐓇𐒰𐓄𐒷 (xin-ha' sha-be. "dark skin"). This word/term is not in their modern dictionary.4
The Lakota dictionary I have access to online isn't as useful, but I see their word for "White" is indeed quite similar to Osage's (Transliterated "ska" instead of çka). So it would probably be quite similar.
So yes, it does look like they had words for both Native Americans as a whole, and for non-white people. However, they were compound words (word-ized phrases), possibly showing their relatively novel nature to their traditional way of thinking.
1 - "Sioux" was apparently an exonym from an enemy tribe, meaning something like "yellow snake". They call themselves "Lakota", so that's generally the preferred way to refer to that people today.
2 - The modern Osage writing system I used here (occasionally anachronistically) is a proper alphabet, mostly using consonant-vowel couplets for each syllable. You can find a pronunciation key on the linked Wikipedia page above, or as a sidebar on the modern Osage dictionary produced by the Osage Nation.
3 - There were also of course some less charitable names, like "yellow eye"
4- As someone well-acquiainted with the intervening era of American History, I can come up with some guesses as to why that term didn't make the modern dictionary. However, a lot of them aren't complementary, so it wouldn't be right to speculate without looking into it. To be fair, it barely made La Fleshe's dictionary, as an alternate form rather than its own entry.