I need more details about the problem. I have not worked with Django before. Note that there is no guarantee that it will work with Django out of the box, maybe you will need to patch Django or the wrapper.
Could it be opening many connections to the same database file? This could be a problem.
Does it keep the connection to the local db open or does it open it only when required and then close it?
Have you discovered which wrapper it uses to interface with SQLite?
Which OS is being used on the nodes?
You can also send me an e-mail with the details you have.