I have been meaning to get around to doing this myself. I have not confirmed this approach will work, but here is my thinking and the limited work I have done.
You need to be using an Editor Add-on (not Drive). From there, you can get a handle on the user's UI.
The following code will pop up a dialogue using the HTML in the CopyHtml.html file, which is another file in my GAS environment.
SpreadsheetApp.getUi().getUi().showDialog(HtmlService.createHtmlOutputFromFile("CopyHtml.html"));
Good luck. Let us know how you get in.