Is it okay to interpolate between overlapping areas obtained using a sliding window?
A first suggestion is to use an odd-sized sliding window (say $w=51$), so that you can really affect the black/white proportion to the center pixel. An even length can generate a half-pixel bias.
And you have a  window sliding by one pixel, therefore you have a black/white proportion for each pixel (except on the borders of the large image, of dimension $W\times W$). The resulting picture will be of size $(W-w+1)^2$, about $1950\times 1950$ here, which already seems to have a large definition for most displays. Hence, I cannot really understand the need to interpolate for visual purposes.
Nevertheless, two contiguous windows share $w\times (w-1)$ (sides) or $(w-1)\times (w-1)$ (diagonal) pixels in common, so more than 98% or 96 % respectively. Hence, the variations in the proportion should vary quite slowly. Then simple interpolation should work nicely. The problem is to give a meaning to the interpolated values. It should be close to having a window with fractional offsets over the big image, and counting partial pixels.