Bootstrap argument to prove regularity of a special solution

The theory for the Dirichlet problem says the solution is unique, and it can be proved that there are at least 2 solutions if $f$ is locally Lipschitz continuous.

No contradiction here. The theory to which you refer (GT Chapter 6) is for the Poisson equation $\Delta u=g$ with given $g$. The equation $\Delta u=f(u)$ may well have multiple solutions, since the right-hand side is different for different $u$.

The normal regularity result that I would use to get this, GT-theorem 8.12, requires that $f\in L^2$.

Indeed, GT do not prove a version of Theorem 8.12 for $1<p<\infty$. But it is true: if $u\in H_0^1(\Omega)$ and $\Delta u=g\in L^p(\Omega)$, then $u\in W^{2,p}(\Omega)$. The proof goes like this:

1. extend $u$ and $g$ by zero to $\mathbb R^n$.
2. Let $v=\mathcal{N}g$, the Newtonian potential of $g$. GT consider it in Chapter 4, but give only Hölder estimates. Sobolev estimates are based on the theory of singular integrals: $D_{ij}v$ is obtained from $g$ via Riesz transforms which are bounded on $L^p$ for $1<p<\infty$. See Theorem 3.2 here, which gives $v\in W^{2,p}(\mathbb R^n)$.
3. Both $u$ and $v$ vanish at infinity, and $u-v$ is weakly harmonic, hence harmonic, hence identically zero.

Also, the last statement by Holder continuity, $u\in C^2(\Omega)$. Why is this?

That book is strangely written. Taking "the integer part of $2-N/r$" is not helpful because this part may well be equal to zero. The point is that the Morrey-Sobolev embedding gives $u\in C^{\alpha}(\overline \Omega)$ for some $\alpha>0$, which implies $g:=u^p$ is in $C^\alpha$. By Corollary 4.14 of GT, $\Delta u=g$ has a solution in $C^{2,\alpha}(\overline \Omega)$. Since the classical solution is also a weak solution, and the weak solution is unique, we conclude that $u \in C^{2,\alpha}(\overline \Omega)$.