You could write a script for whenever the user clicks on the delete button. The script could be similar to below:
Set Error Capture [On]
Allow User Abort[Off]
Re-Login[Account Name: Get(AccountName)]
Delete Record/Request [No dialog]
This way, if they get their password correct, it will delete the record. If they get it wrong or cancel, it will exit the script.
With Filemaker Advanced, you can set up the layout with a custom menu that does not have Delete Record as an option. You can also set up the custom menu so that Delete Record ( and Ctrl-E ) perform a delete record script that you devise. You can also remove the keyboard shortcut key combo in the custom menu as well.
Without Advanced, you can set up the users with a privilege set that does not permit deleting records. You can then use a button on the layout for deleting records that performs your delete record script and that is set to "run with full access privileges".