Probably. Yes. Maybe.

The factors that would make what you describe feasible are mostly related to timing and location. The difference in American education between 1820 and 1880 was dramatic in terms of bureaucracy, laws, accessibility, cost, and feasibility. Likewise, the education structures in Texas following the Civil War were different than they were in Dakota Territory before the war.

That said, it was basically a chicken and egg situation. If a multi-racial family moved to a particular area with plans of having children and setting up a community around other people, they likely already had a plan for education. That is, some families brought libraries with them or others sent for a schoolteacher once a schoolhouse was built - or someone had enough room in their house for the schoolteacher to sleep and a room for her to hold class. Moving west to educate children was seen as a noble and righteous option for young (mostly white) Protestant women. The young women who left the East Coast in the 1830 to 1870s or so to teach in newly formed communities out West had a limited degree of autonomy and it's possible that a teacher could refuse to teach in a schoolhouse if there was an identifiably Black child but that would, in effect, mean she failed at the thing she was meant to do. On the other hand, there were plenty of young women who went West to teach - especially Quakers - who likely would have been nonplussed by the presence of non-white or multiracial children. The young women were expected to teach no matter where they placed and deal with the problems they faced, regardless of context. To a certain extent, it served as a missionary experience for her. School had a distinctly Protestant bent and while she couldn't serve her church, she could serve children. So, refusing to serve because a child was non-White could have conflicted with her responsibilities as a teacher.

If a multi-racial couple didn't have a plan and suddenly found themselves with a child, unsure of their educational options, a great deal would come down to their wants and needs for their child. School bureaucracy was fairly limited before the 1900s outside the New England states which meant there wasn't likely a schoolman checking to make sure all of the children in a schoolhouse were white. If a multi-racial child showed up at a frontier schoolhouse, there was no reason for the teacher to ask for their race or identity and no reason to exclude a child on account of their race. This isn't to say all teachers were welcoming to non-white children, but rather, attendance was so sparse, refusing to educate one child who did show up would, again, stand in conflict with her responsibility as a teacher. Integrated schools weren't illegal per se on the Frontier (laws against integrated schools were mostly found in Southern schools in the 19th century and while Brown v. Board primarily involved schools in the mid-west, those schools and districts weren't built until well into the 1900s.)

All of which is to say, education on the frontier was highly idiosyncratic. There are examples of communities established by previously enslaved Black families who built a schoolhouse as soon as it was feasible. There are examples of European or Russian families who settled in the west territories and used the presence of a schoolhouse as a lure to bring others from their community and supported bilingual education and others who insisted on English-only instruction in the school. Finally, there were children who had European and Indigenous parentage who had vastly different experiences. Some wee went to white schools with the goal of assimliating into white culture, while others went with the goal of returning to their Indigenous community and acting as a go-between and translator.

So, basically, probably. Yes. Maybe.