If I were you, I wouldn't fill it up too full and I would dump it every day. Kids tend to urinate in it; bring in dirt; grass, sunscreen, mosquito repellent, pets, toys, etc. Yes, you are right, mosquito larvae don't take long to hatch. I wouldn't put anything in the water for insects that a child could ingest, unless it is harmless to people & pets. I am a swimmer and I swallow some pool water almost every time. Kids tend to drink pool water more often. Algae are also an issue in water, and grows with the sun. To keep the water fresh and clean, dump & re-fill every day--this is the "natural" filter system.
As for your beauty rest: I recently read a research study on the Net about 8 hours sleep per night for men vs. women. Guess who tends to get diabetes and heart disease as a result of insufficient sleep? You guessed it: women! You can try to nap when your child is napping.
Good luck! J.