I would bet you could buy an insert that you could screw a commercial stabilizer into that could be mounted in the riser section of a bow. I've seen fiberglass recurves with them.
Don't worry this is an open minded group of people, you should have no problem finding some info.
A stabilizer might not be something you would find on a traditional bow, but neither is a broom handle take down or a fiberglass or pvc sleave take down. I'm also fairly certain traditional bowers didn't use super glue or even epoxy glues. Everyone needs to determine how primative they want to be and where they draw the line.
Let he who does not use a chainsaw for cutting his staves, a car for transporting them, a bandsaw for shaping them or a climate controlled electrically lit building for storing and drying them throw the first metephoric stone.
I would bet that my ancestors both native american and european would have used a stabilizer on their bows if it would have helped there accuracy in the hunt or war.
Don't let other people dictate the direction you want to try. I can imagine several ways to create and mount a "primative" stabilizer. The art and craft of wooden bows should not be frozen in past desings simply because someone else says they should. And if that is the limit of this forum I would like to see the historic models that the last years bows of the month were based on.
As for the utility of some sort of stabilizer on such a bow, based on the comments not sure you will find a definative answer here. I say give it a try. Worst case you tweek a few people's ego's when you screw the stabilizer into your osage self bow.