Thanks for your comments everyone! It was not a sewer pump as it turns out (we checked), just a regular one.
Even after the last big rainstorm, it doesn't seem to be changing the water in the bottom, is there a setting on the pump which would change how far down it pumps out?
We put bleach in the water, and noticed an immediate improvement in the smell. Will bleach cause any parts to wear (rubber seals or anything?) over the long term, if we just keep putting bleach down every few weeks? We don't generally have oxy clean or peroxide around.