Option 1 -> sullenly means in a bad-tempered manner, which is similar to churlishly, not opposite.
Option 2 -> unpleasantly means in a disagreeable way, which is similar to churlishly, not opposite.
Option 3 -> rudely means in an impolite manner, which is similar to churlishly, not opposite.
Option 4 -> politely means in a courteous and respectful manner, which is the direct opposite of churlishly.
Hence, Option 4: politely -> "Churlishly" means behaving in a rude, bad-tempered, or ungracious way. The opposite would be acting with courtesy and good manners, which is captured by "politely." -> correct