You cannot just believe something like that. You need to rigorously justify it. For now it's wrong. As for the trick itself, this is what I mean.

You are taking the laplacian of a sum that does not converge absolutely(EDIT: uniformly)on the range 1/2 <Re(z) < 1. I don't think the trick could work because you are doing 2nd order derivatives on the function and THEN you are adding up the results. First you need to show absolute(EDIT: uniformly)convergence for the series for the given range so that you can change the order of differentiation and then once you find the 1st order derivatives, you need to assure absolute(EDIT: uniform)convergence for the results on the interval so that you can change the order again. The trick you are talking about comes after you have made a total of 4 illegal changes of order. It is very unlikely to work.

The Dirichlet series of Zeta is derived by adding two series, yes. However, you are hoping that differentiating would do the trick? I highly doubt it.