The error in your derivation is that the equation you use for dE/dV is only valid at constant temperature, see

https://physics.stackexchange.com/questions/336012/thermodynamics-what-is-de-dv-t

For that reason, if you want to integrate over temperature, thus making the constant T a variable, the formula gets an additional term which would be expected to compensate for the missing summands in your result.