Hmm, you could rewrite it this way: (I'm going to assume that $\pi_0, \pi_1, \xi'', \mu, \lambda$ are pre-defined constants, and $\alpha,W$ are variables)
$$ \begin{align} &\max_{W}\; \pi_{0} (1 - e^{-\mu W}) - \frac{\pi_{1}}{W}z_{1}(W)\\ s.t.\;& \frac{dz_{0}(\alpha)}{d\alpha} = 1 - e^{-\mu \alpha},\quad z_{0}(0) = 0\\ & \frac{dz_{1}(\alpha)}{d\alpha} = 1 - e^{-\lambda \alpha},\quad z_{1}(0) = 0\\ & z_{0}(W) + \epsilon \leq \xi''\\ & W \geq 0 \end{align} $$ where $z_{0},z_{1}$ are auxiliary variables and $\epsilon$ is a numerical tolerance value. This then becomes a DAE (differential-algebraic equation) optimization problem. I don't believe there exists an analytical solution for this problem, owing to the algebraic inequality constraint (otherwise one could apply Pontrayagin's maximum principle) -- but it is certainly possible to solve this system numerically.
Mind you, this is a nonconvex optimization problem, so a global solution is not guaranteed if you apply local methods.