Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$.  Certainly 
$$
\text{Pic}(X, \pi_* O_{\widetilde X}) \cong \text{Pic}(\widetilde X, O_{\widetilde X})).
$$
Also observe that $(\pi_* O_{\widetilde X})^{\*} = \pi_* (O_{\widetilde X}^{\*})$.  Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that 
$$
\text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*).
$$
Combining with the isomorphisms already written completes the proof.