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Pullbacks regard maps between manifolds, while transition maps are maps between chart codomains. Maybe it is a matter of semantic, but it seems to me that you are interested in having the expression of the differential form tauin terms of spherical coordinates, which is not what I would call a pullback: a pullback yields a different differential form, while changing coordinates simply yields a different component expression of the same differential form. From your example, we have

sage: tau.display()  # expression in the default chart (Cartesian coordinates)
tau = dx/\dy/\dz
sage: tau.display(spherical.frame(), spherical)
tau = rho^2*sin(theta) drho/\dtheta/\dphi

Is the last expression what you were looking for?