I stumbled across the book Second Oder Elliptic Equations and Elliptic Systems by Yah-Ze Chen which appears to contain an answer to my question. It is available on google books here
http://books.google.com/books?id=eQcbiPQPweQC&pg=PA49&dq=strong+solution+dirichlet+problem+Lp&hl=en&ei=byKiTeXXO6GG0QGG7dGgBQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCcQ6AEwADgK#v=onepage&q=strong%20solution%20dirichlet%20problem%20Lp&f=false
To save time for those who are interested, here is the relevant argument:
For large $\lambda > 0$ we want to show that $L_{\lambda} = L - \lambda I$ is injective on $W_0^{1,p}$.
Claim: Let $L^T_{\lambda}$ be the transpose of $L^{\lambda}$ with respect to the paring that defines weak solutions. Then we claim that $L^T_{\lambda}$ inective on $W_0^{2,p}$ implies that $L_{\lambda}$ is injective on $W_0^{1,p}$
Proof: Suppose that $L^T_{\lambda}$ is injective on $W_0^{2,p}$. Then, by an argument contained in the original post above, for every $f \in L^p(\Omega)$ we can find $u \in W_0^{2,p}(\Omega)$ such that $L^T_{\lambda}u = f$. Now, suppose that $L_{\lambda}v = 0$ for some $v \in W_0^{1,p}$. After an integration by parts and the definition of weak solution, we see that $\varphi \in W_0^{2,q}$ implies that
$\int_{\Omega}uL^T_{\lambda}\varphi = 0$.
Now choose $\Omega'' \subset\subset \Omega' \subset\subset \Omega$ and a bump function $\rho$ identically one in $\Omega''$ with support in $\Omega'$. $\rho\text{sgn}(u)$ is in $L^q$, and we can find $g \in W_0^{2,q} $ such that $L^T_{\lambda}g = \rho\text{sgn}(u)$. Plugging this $g$ into the above equality gives
$\int_{\Omega''}|u| = -\int_{\Omega\setminus\Omega''}\rho |u|$
Due to the arbitrariness of $\Omega''$, this implies that $\int_{\Omega} |u| = 0$ and hence $u$ is $0$ a.e.
Claim: For $\lambda$ large enough, $L_{\lambda}$ is injective on $W_0^{2,p}$.
Proof: Suppose $L_{\lambda}u = 0$ for $u \in W_0^{2,p}$. Let $\tilde{\Omega} = \Omega \times (-1,1)$, and $\tilde{\Omega'} = \Omega \times (-1/2,1/2)$. Let $(x,t)$ be the coordinates on $\Omega \times (-1,1)$. Then define $v(x,t) = \cos(\sqrt{\lambda}t)u(x)$. Let $\hat{L_{\lambda}} = L_{\lambda} + \partial_t^2$. We have $\hat{L_{\lambda}}v = 0$. The strong solution estimates give
$\vert\vert v\vert\vert_{W^{2,p}(\tilde{\Omega'})} \leq C\vert\vert v\vert\vert_{L^p(\tilde{\Omega})} \leq C\vert\vert u\vert\vert_{L^p(\Omega)} \Rightarrow $
$\vert\vert \partial_t^2v\vert\vert_{L^p(\tilde{\Omega'})} \leq C\vert\vert u\vert\vert_{L^p(\Omega)} \Rightarrow $
$\lambda\vert\vert u\vert\vert_{L^p(\Omega)}(\int_{-1/2}^{1/2}|\cos(\sqrt{\lambda}t|^p)^{1/p} \leq C\vert\vert u \vert\vert_{L^p(\Omega)} \Rightarrow$
$\lambda^{1 - \frac{1}{2p}}\vert\vert u\vert\vert_{L^p(\Omega)}(\int_{-1/2}^{1/2}|\cos t|^p)^{1/p} \leq C\vert\vert u\vert\vert_{L^p(\Omega)}$
Now taking $\lambda$ large enough implies that $u = 0$ almost everywhere.
The paper Besov Regularity for Elliptic Boundary Value Problems by Dahlke and DeVore discusses regularity results for these problems on Lipschitz domains. I reproduce their theorem 4.1 below:
Theorem 4.1 Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^d$. Then, there is an $0<\epsilon < 1$ depending only on the Lipschitz character of $\Omega$ such that whenever $u$ is a solution to
$$\begin{align} - \Delta u &= f ~ on ~ \Omega \subset \mathbb{R}^d, \\ u &= 0 ~ on ~ \partial \Omega \end{align}$$
with $f \in B_p^{\lambda-2}(L_p(\Omega))$, $\lambda:=\frac{d}{d-1} (1+\frac{1}{p})$, $1 < p<2+\epsilon$, then $u \in B_\tau^\alpha(L_\tau(\Omega))$, $\tau=(\alpha/d+1/p)^{-1}$, for all $0 < \alpha < \lambda$.
The proofs therein use an interesting technique of expanding the solution via a wavelet multiresolution analysis, and then using the connection between the decay rate of a functions wavelet coefficients and the smoothness space that function lies in. It turns out that Besov spaces are the proper smoothness spaces for elliptic PDE solutions on "bad" domains, or with rough boundary data.
For some background on Besov spaces and wavelets, see the book Theory of Function Spaces II by Triebel, and the excellent expository paper Wavelets by DeVore and Lucier.
A more classical treatment is given by Grisvard in his book, Elliptic Problems in Nonsmooth Domains, though the results are not quite as good as the newer Besov results.
Best Answer
While probably not the fastest approach I think that Hörmander: The analysis of linear partial differential equations, IX:thm 9.5.1 seems to give a (positive) answer to your question. It is overkill in the sense that it gives you a microlocal statement telling you that for $Pu=f$, $u$ is analytic in the same directions as $f$ is.