I did not manage to actually call the doGet() method. However, I am now filling the plugin’s space admin page’s fields on load via several getFieldName() methods in the Action class. Not exactly what I wanted to do but it does get the job done.
For completion’s sake, this is what the methods in the MySpaceAdminAction class look like:
public String getFieldName() {
return MyConfluencePluginSettingsHandler.getFieldName(this.getSpaceKey());
}