I was able to reproduce this and it seems that no combination of permissions is allowing me to access the fields. I will try a few more possibilities and discuss it with the developers but in the meantime you can remove the permissions check under ~\CMS\CMSModules\Bizforms\Tools\BizForm_Edit_Data_SelectFields.aspx.cs
By commenting out lines 55-58:
if (!user.IsAuthorizedPerUIElement("CMS", "CMSDesk.Content"))
I will provide an update once I have more information.