Why do you want to fill the whole map with white? If everything is going to be equally opaque you don't need a transmap at all. If you want to be selective, assuming your paint program supports layers, copy the grid to a new layer and set the blending mode to multiply, then clear and paint on the background - the black will be opaque but the white will show through as you work, and you can hide the grid layer from time to time to check progress. You should have some sort of polygon lasso tool, so if you need to flood fill a large area click around with that using the grid as a guide (you don't have to follow the inner lines if you don't want to). When the texture is done hide/delete the grid layer and save as jpg.