Is Kirchoffs law valid at all pressures?
It would say that you are making three separate mistakes:  


The standard state pressure is any arbitrary pressure, not necessarily 1 bar, so there is no implication from the equation specifying standard state that it is only true at one particular pressure.  
The important specification of being in the standard state is not the value of arbitrary pressure chosen, but that the substances are not mixed.  They are pure or in solvent as the single solute.  Also, for gases and solutions the states are fictitious.  In your derivation, you are not accounting for changes due to mixing and any changes due to being in real, rather than fictitious, states.   
I don't see why you say to integrate Cv rather than Cp.