Maybe a possibility would be to first get rid of the non-linearity by defining $\phi[\tilde{p}]$ to be the solution of :
$$
\begin{cases}
p_t = e^{-\tilde{p}}p_{xx}, & (t,x)\in (0,T)\times (0,1),\\
p(0,\cdot)\equiv -c &\text{for }x\in (0,1)\\
p(\cdot,0)\equiv 0 \equiv p(\cdot,1) &\text{for }t\in (0,T),
\end{cases}
$$
and then try to find a $p$ that satisfies $\phi[p]=p$ by some fixed-point arguments.

For one, if $p$ was a continuous solution of the original problem, then by the comparaison principle, $-c\leq p\leq 0$. So the space to study could be the space of continuous functions such that $p(\cdot,0)=p(\cdot,1)=0$ and $-c\leq p\leq 0$.

In that case, $e^{-\tilde{p}}$ would be very well bounded and behaved, so all the usual existence and regularity results apply (section 7.1 of "Partial Differential Equations" by Evans can pretty much be applied directly).
To me, the main difficulty from there is proving the continuity of $\phi$ or similar properties to apply fixed-point theorems. I don't know of many estimates that explicitly involve the diffusion coefficients.