Actually, there are a number of mythological spiders that could be used. Egyptian goddess Neith, Greek's Arachne (as well as Athena, to some extent), African's Anansi (also known as Kwaku Ananse), Lakota's Iktomi, Navajo's Spider Grandmother, Japan's Tsuchigumo and Jorogumo. Anansi the trickster or Arachne are probably the most likely to be used.
For bats, I couldn't find any specific mythological creatures/deities dealing with them, but given that they're traditionally associated with vampires, we could see something based on vampires.