Elliptic regularity in Hölder spaces is precisely what you need here.

If you want to prove it along the lines you described, using $L^2$-theory, then you can first establish that smooth $f$ leads to smooth solution $u$. Then you approximate $f$ by smooth functions (in the $C^{\alpha}$-norm), and use the Schauder estimate. It suffices to do everything locally.

Elliptic regularity in Hölder spaces is precisely what you need here.

If you want to prove it along the lines you described, using $L^2$-theory, then you can first establish that smooth $f$ leads to smooth solution $u$. Then you approximate $f$ in the $L^2$-norm by smooth functions with uniformly bounded $C^{\alpha}$-norms (this is important as smooth functions are not dense in Hölder spaces), and use the Schauder estimate. It suffices to do everything locally.