This might be useful: Why no refresh_data_bindings() for write back data binding? - #25 It uses the Anvil Extras library, and adds in a way to trigger the writeback programmatically. Not sure if it works for date pickers or not.
Another option would be to set the self.item value instead of the date picker value. You have data binding in place, so the date picker would be updated automatically when you update the corresponding self.item entry.