You could intercept IContentCopyHandler and pass in false for parameter "publishOnDestination" as:

 [ModuleDependency(typeof(EPiServer.Web.InitializationModule))]
    public class CheckedOutCopyInitializableModule : IConfigurableModule
    {
        public void ConfigureContainer(ServiceConfigurationContext context)
        {
            context.Services.Intercept<IContentCopyHandler>((locator, defaultCopyHandler) => new CheckedOutContentCopyHandler(defaultCopyHandler));
        }

        public void Initialize(InitializationEngine context)
        {}

        public void Uninitialize(InitializationEngine context)
        {}
    }

    public class CheckedOutContentCopyHandler : IContentCopyHandler
    {
        private readonly IContentCopyHandler _defaultHandler;

        public CheckedOutContentCopyHandler(IContentCopyHandler defaultHandler)
        {
            _defaultHandler = defaultHandler;
        }

        public ContentReference Copy(ContentReference contentLink, ContentReference destinationLink, AccessLevel requiredSourceAccess, bool publishOnDestination)
        {
            return _defaultHandler.Copy(contentLink, destinationLink, requiredSourceAccess, false);
        }
    }