Is this Laplace BVP well posed? If not, why?
Special solutions of the Laplace equation that obey all of the OP's boundary conditions are:
$$
\phi_k(x,y)=e^{-k\pi/a.y}\cos\left(k\frac{\pi}{a}x\right) \quad \mbox{with} \; k=1,2,3,\cdots
$$
The Laplace equation is linear, hence a more general solution is:
$$
\phi(x,y) = \sum_{k=1}^n c_k\,\phi_k(x,y) \quad \mbox{with} \\
\phi(x,0) = \frac{1}{2}c_0 + \sum_{k=1}^n c_k \cos\left(k\frac{\pi}{a}x\right)
$$
The latter expression can be interpreted as a (partial) Fourier expansion of the even periodic continuation of $\,\phi(x,0)=f(x)\,$:
$$
c_k = \frac{2}{a} \int_0^a f(x)\,\cos\left(k\frac{\pi}{a}x\right)\,dx
$$
In particular:
$$
\frac{1}{2}c_0 = \frac{1}{a} \int_0^a f(x)\,dx = 0
$$
Previous answer (heuristics) deleted