Is it meaningful to assume static pressure to be constant along a streamline?
If you assume both hydrostatic and pure geostrophic balance, that is a valid assumption.

In Einstein notation, $$u_i=-\frac{1}{f \rho}\frac{\partial P}{\partial x_j}\epsilon_{ij3}$$

If we look at the equation for the streamline:$$u_i=-\frac{\partial \psi}{\partial x_j}\epsilon_{ij3}$$, then we can see that 

$$-\frac{\partial \psi}{\partial x_j}\epsilon_{ij3}=-\frac{1}{f \rho}\frac{\partial P}{\partial x_j}\epsilon_{ij3}$$

which simplifies down to $$\frac{\partial \psi}{\partial x_j}=\frac{1}{f \rho}\frac{\partial P}{\partial x_j}$$. Thus, for isochoric (incompressible) geostrophic flow on an f-plane, the streamfunction only depends solely on pressure.