While it's possible that your father wasn't recorded in the 1920 census, there are a lot of avenues of research to follow which might let you find where he was recorded.
First, since he's your father you know he was definitely alive in 1920.
Second, you know that his family was split up in 1920, with his father living in one place and his mother in another place. That certainly suggests that his family had some personal or financial reason to be living apart, and makes it much more likely that your father was living somewhere else, rather than just having been forgotten.
Third, you know that your father was 10 years old in 1920, and so almost certainly wasn't living on his own.
Given that your father was alive in 1920, and it is quite plausible that he wasn't living with either parent, and likely living with one or more adults who were taking care of him, where might he have lived?
There are a few possibilities:
- He might have lived with relatives. The most likely relatives are grandparents, aunts or uncles. If you locate the 1920 census records for all of those relatives, you might find a 10-year-old boy listed as grandson or nephew living with them. Keep in mind that not all of those relatives will have the last name Taber. The census record might not list his last name, in which case it will have been indexed under his relative's last name.
- He might have lived in some kind of institution. For instance, maybe he was at a boarding school, or an orphanage, or a reform school. I have seen records of children who were placed at an orphanage even though both parents were still living.
- He might have lived with an unrelated family, as a servant or apprentice or something like that.
- You say your father was part Cherokee. Perhaps he lived on a Cherokee reservation in 1920, either with close relatives, or with distant relatives or friends of his parents.
- You say your father had relatives in Muldrow, Oklahoma. I see that it is only 12 miles from Fort Smith, Arkansas, where he was born, so it is very possible that he might have lived in Muldrow in 1920. Fortunately, the population of Muldrow was only 693 in the 1920 census, so you should be able to read the entire list of people who lived there in the 1920 census record images, and see if anyone is likely to be your father. You might find that the last name is wrong, but the age and first name or other details are a strong enough match to conclude it was him.
Putting it all together, the research plan I would suggest is to figure out the names and relationships of as many of close his relatives as possible, and then look for those people on the 1920 census; and also consider institutions he might have lived at. You might find him listed in one of those places.
Edit: This page (and two follow-up pages it links to have a number of interesting search suggestions when looking for a person's census record: https://www.rootstech.org/blog/8-strategies-for-finding-missing-persons-in-the-us-federal-census