What are you changing in the twbx file ?
Do you know that it's "only" a zip file with a twb, tde and a few other files contained ?
It sounds like you might be better off using Tableau Server and a published data source to remove all the manual effort involved.
Each twbx will include the data of a specific client (since clients can't see each other's data).
Tableau Server is the way to go on this, I agree. I just had to explore any other options due to budgetary constraints but this is the only way to do it.
If it's only the data that's changing each month, then possibly using Alteryx (or similar) to create a TDE file and then only send the updated data to your clients might be an option.