Hi Micha,
The designer files are generated by visual studio. You don't add things to the designer files, you add things to webform controls (ascx) or pages (aspx) and visual studio adds a reference to the class which defines the things (like buttons) you add to the controls or pages.
If you want to add a button which will initiate a sync with an external database, you could add it to the settings of the module. See this documentation for more information.
Thanks,
Joe