This is probably not unique. Due to translational invariance, we can restrict the space on which we're acting to functions $f$ on the interval $[-1/2,1/2]$, and if we specify those to be odd and antiperiodic, $-\frac{1}{2} \frac{d^2 }{dx^2} $ seems a viable inverse:
$$
-\frac{1}{2} \frac{d^2 }{dx^2} K(x,y) = \delta (x-y)
$$
If we act with $K$ on an $f$ from our space, we again obtain a result from our space,
$$
\int_{-1/2}^{1/2} dy K(1/2,y) f(y) =
\int_{-1/2}^{1/2} dy \left[ 1-\left| \frac{1}{2} - y \right| \right] f(y) = \int_{-1/2}^{1/2} dy \left[ \frac{1}{2} + y \right] f(y) = \int_{-1/2}^{1/2} dy y f(y) = -\int_{-1/2}^{1/2} dy [-y] f(y) = -\int_{-1/2}^{1/2} dy \left[ \frac{1}{2} - y \right] f(y) =  -\int_{-1/2}^{1/2} dy \left[ 1-\left| -\frac{1}{2} - y \right| \right] f(y) = -\int_{-1/2}^{1/2} dy K(-1/2,y) f(y)
$$
(antiperiodicity), as well as
$$
\int_{-1/2}^{1/2} dy K(0,y) f(y) = \int_{-1/2}^{1/2} dy \left[ 1 - |y| \right] f(y) = 0
$$
(oddness). Of course, the same is true for acting with the inverse.