Thanks to intervention from Microsoft (definitely worth the flat fee they charge per incident), we were able to identify the problem. Instead of using the network path to identify the website location on the "Home Directory" tab of the IIS properties, we were using the local drive path. That was all that needed to be changed.
Once we switched to the network path and added a dedicated service account to "Connect As...", impersonation started working right away. Users pass their logged on credentials via integrated authentication (no logon required) and the service account takes care of executing their actions on the database file.
Access to the shared folder is limited to a brief list of administrators, and data access on the web application is limited based on user names.
If anyone is stuck with this and needs help, let me know!