If it is only about the connection string, add a reference to a second config file which you deploy to your folder, like:
<connectionStrings configSource="connectionstrings.config" />
Or deploy your complete config file seperately, and load it in your application like:
var map = new ExeConfigurationFileMap {ExeConfigFilename = @"C:\MyProject\custom.config"};
Configuration config = ConfigurationManager.OpenMappedExeConfiguration(map, ConfigurationUserLevel.None);