Its looks like one solution to this is the function $$f(w,x) = 1+\frac{1}{2\pi i} \int_{\frac{1}{2} - i \infty}^{\frac{1}{2} + i \infty} w^{t^2} (-x)^t \Gamma(-t) dt $$ Which is obtained by applying the residue theorem and assuming the other parts of the contour cancel out somehow. This solutions agrees with the approximations provided by the previous function, though this one converges well enough to be able to numerically confirm that it satisfies the solution to the delay differential equations. It has a branch cut at $\theta = -\pi$, so for certain angles of $w$ and $x$ the differential equation needs to evaluate the function at different branches to satisfy the differential equations. We can obtain all of the branches as $$f_n(w,x) = 1+\frac{1}{2\pi i} \int_{\frac{1}{2} - i \infty}^{\frac{1}{2} + i \infty} w^{t^2} e^{ \left(\ln(-x)+2\pi i n\right)t} \Gamma(-t) dt $$
What is left is prove uniqueness. This solution is interesting to me because the function $f(w,x)$ when $|w|<1$ is a complex function with a natural boundary, and so if we can prove uniqueness of the solution for $f(w,x), |w|>1$ it may provide an interesting way to 'analytically continue' the inner function beyond its natural boundary.