See if this helps:
1. Publishing your data source to the Development Server.
2. For the workbook, switch the data source to the one you have just published above.
3. Publish this workbook to the Production Server. Hopefully, it will work, unless your production server cannot connect to the Development Server.
You can do that. The current process requires a connection to your laptop because that is how the data source is defined in the Tableau Workbook.
The "easiest" way would be to extract the .tde file from an extracted version of the current workbook (right-click and unpackage in Windows Explorer).
Then, add a new data source to your workbook that uses the .tde file rather than the live connection. Start with a "More..." file type connection:
Navigate to the location of the .tde file:
IMPORTANT: Make sure the location of the file is somewhere that is accessible from the server and your laptop. Otherwise, it will not be able to be updated.
Once you have the new data source, replace the existing data data source (data > replace data source):
Close the current data source.
Save the new workbook as a different name
You can create a new .tde file every time the data gets updated and place it in the shared location.
There are various ways to create .tde files (manually or using the API). After that, the process will be the same.