At some point, any GA computation bottoms out to scalars. Since you start out with quantities you already know how to give dimensions to, I would expect you can

At some point, any GA computation bottoms out to scalars. Since you start out with quantities you already know how to give dimensions to, I would expect you can just follow standard dimensionsal analysis. That isn't to say it's straightforward to implement computationally, or that there isn't interesting structure there.