Better Exposed Filters and a Node Relationship
I don't think this is related to the relationships. It looks like Drupal core date fields (post, changed, etc.) are not getting the proper BEF options, namely the datepicker. That's a bug in BEF -- it was mentioned at the end of this issue, but I never saw that because that issue is closed. Feel free to post a bug in the BEF issue queue for this.

Thanks.