Is there a way to disable export options for public links on charts, dashboards, etc?
I don’t see a setting for that, no. Of course, you’ve already sent the client the data, is it a big deal if you send it to them again? 
If someone wanted the data even with no export button, they could get it from the browser developer tools, either console or network activity.
Also note, the docs don’t recommend using public sharing for questions that protect private data via a filter as the filter can be trivially bypassed.