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If your library is being included on a webpage without bunding - that is, with
<script src="..."></script>
<script>//other client script</script>
then the most sensible approach by far (the one that the vast majority of these sorts of libraries implement) is to create an object that's accessible globally that provides an interface for all your library properties and methods. Dispatching a click event to the script tag doesn't make sense. Rather, have your library do something like:
window.theLibrary = (() => {
const theLibraryNamespace = {};
// lots and lots and lots of code that assigns stuff to theLibraryNamespace
return theLibraryNamespace;
})();
Then, to interface with it, the other scripts on the page would do something like
theLibrary.sendMessage('someEventName', eventData);
or maybe
theLibrary.doX(eventDataForX)
(with whatever sort of function signature(s) fits your expected use case best)