Hi Catharine,
Are there any errors in the Qdabra or Application logs? I imagine there is a possibility that DBXL is failing to map using Web Service. With Object Model (and Publish Library and XSN checked) we have another way of mapping the form from DBXL to SharePoint. However, sounds like this may be failing in your environment. (409) Conflict is not an error I am familiar with but perhaps the event viewer has additional details, or perhaps switching to Object Model solves the issue.
One final resort might be to:
1. Modify your submit data connection such that the doctypename is hardcoded (pgs 1-2 in this document)
2. Publish the form from InfoPath directly to the SharePoint form library.
3. Then attach the template.xsn (which is in SharePoint) to DAT. That is, instead of attaching a local copy of the XSN, attach the copy that is stored in the SharePoint form library, i.e. http://<servername>/<YourFormLibrary>/template.xsn
This way DBXL is forced to use the form template stored in SharePoint, and we force SharePoint to have the latest via the InfoPath publishing wizard. Make sense? Again, this is an option I would suggest trying if the SharePoint mapping woes cannot be resolved.
DBXL v2.3 was the first version of DBXL with SharePoint mapping, and improvements have been made since then, so perhaps another option is to upgrade.
Let us know if you need additional information. Thank you.