Is it possible to make a script that will, so to say, "save as" the current sheet to a new name, making a copy, that can then be immediately opened in a new tab? The programmatic equivalent of File->Make a copy.
You can definitely generate a copy, store it in a defined folder, and display a link that when clicked will open in a new window.
function copyWorkbook() {
var spreadsheet = SpreadsheetApp.getActiveSpreadsheet();
var name = spreadsheet.getName();
var spreadsheetId = spreadsheet.getId();
var todaysDate = new Date().toLocaleString();
var destFolder = DriveApp.getFolderById("destination_folder_sheet_ID");
var copyFile = DriveApp.getFileById(spreadsheetId).makeCopy(name + ' ' + todaysDate, destFolder);
var copyId = copyFile.getId();
var ui = SpreadsheetApp.getUi();
var url = `<a href="https://docs.google.com/spreadsheets/d/${copyId}/" target="_blank">${name}</a>`;
var html = HtmlService.createHtmlOutput(url);
SpreadsheetApp.getUi().showModalDialog(html, "testing");
}