Converting a polygon to pixels is something that graphics libraries are highly optimized at. So I'd go with the graphics library approach.
It may be as simple as drawing a border around each polygon in white. If your lookup finds a white pixel, then look at nearby pixels to find the colors of polygons to test the 'long' way. Then you don't have to have a really high-resolution image.
If you've got very thin strips, then that may require special handling. Assigning one more color to each polygon that has such should cover it, though (that color means try that polygon but also try polygons represented by the color of nearby pixels).
An alternate approach would simply be to draw the polygons without borders (being sure to draw any really thin strips last) and always look at the 9 closest pixels instead of just 1 pixel.
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