I would say it is practical and reasonable to use the format that is used on rosters and/or the service records or to chose what appears to be the most common for the person. For example, you will find the same person as "Coming Deer" and "Deer Coming". "Coming Deer" is the most common usage.
There is much variety in how Indian names were written or translated to English. You almost have to consider each name individually and use your best judgement. In researching Cherokee records, it is interesting to see how how names and surnames were created and evolved over time.
This evolution is very different between those who were and were not educated in English. That is, those educated in English 'established' a standard form for their surnames at an early date -- for example, the son of Oo-wa-ti (the ancient one) was named De-ga-do-ga (standing together). De-ga-do-ga was educated in English and permanently established his own name as "Stand Watie".
The names you are currently working with are generally Indians who were not educated in English. The Southern Indians were more often educated in English and their names have more consistent structure.
The English translation should be considered as if one 'word'. ("Chickasaw Killer" should NEVER be "Killer, Chickasaw", though you will run across that in the rosters and rolls on occassion.)
Longer phrases are often hyphenated with only the first word capitalized, e.g., "Deer-in-the-water".
Two word names, e.g., "Chickasaw Killer" and "Coming Deer", are usually not hyphenated and both words are capitalized. In some cases, the words are combined into one word, e.g., Tenkiller, Mankiller, Hogshooter. The combining of names this way is NOT consistent in the records of the period, meaning you will see "Hog Shooter" and "Hogshooter" used for the same person, even in the same service record.
These translations of the native language name to English are most commonly listed as surnames with no first name. (They technically were not "surnames" but were adapted to surnames later -- that is, the structure of surnames and first names comes from the Europeans and is not traditional to these tribes.) Consider again Oo-wa-ti (the ancient one) who the missionaries began calling "David" and then "David O'Watie" to distinguish him from other "Davids" and later "David Watie". His name was just "Oo-wa-ti" -- which was not a surname/family name. The missionaries introduced the first name and surname structure which was carried to his sons Stand and Buck Watie (aka Elias C. Boudinot in honor of a benefactor).
Star Deer-in-the-water was (I think) the son of Deer-in-water. In his case, the surname is, of course, "Deer-in-the-water" and the first name is "Star". Star was killed by Watie's regiment -- as I remember, in early 1863, he was at home on either Greenleaf Creek or the Illinois River when a scout from the 1st Cherokee Mtd Vols came up from their camp a Webbers Falls. (Just mention that 'cause I remembered you did some training at Camp Gruber which incompasses some of Greenleaf Creek).
In cases where there is an adjective in English with a native language name, the English adjective is shown as the first name. For example: for Skiatooka, Big Skiatooka, and Young Skiatooka, you would list the names as "Skiatooka" (as a surname with no first name), and then "Skiatooka, Big", and "Skiatooka, Young".
When the surname is in English, you have to use your judgement on whether the adjective is part of the native language name or added later to the English name. For example, if the native name translated to "Big Mush", then the surname is "Big Mush". BUT, if the native name translates to "Mush" and they call his son "Little Mush" and call the father "Big Mush" then the surname is "Mush" and the first names are "Little" and "Big". Often you just can't tell without a lot of digging.
An awful long explanation for a short answer -- go with what is most common in the rosters/service records.