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.