My cat likes to jump on the bathroom counter to lap drops of water out of the sink, so using harsh chemicals to unclog the drain is not an option.

Make your own harmless and environmentally friendly drain cleaner with baking soda, vinegar, and boiling water.

eartheasy has the recipe:

Drain Cleaner: Pour about 1/2 cup baking soda down the drain, then 1/2 cup vinegar. The resulting chemical reaction can break fatty acids down into soap and glycerine, allowing the clog to wash down the drain. After 15 minutes, pour in boiling water to clear residue. Caution: only use this method with metal plumbing. Plastic pipes can melt if excess boiling water is used. Also, do not use this method after trying a commercial drain opener–the vinegar can react with the drain opener to create dangerous fumes.