To start a presentation, you have to make your script to open the presentaion in presentation mode. To do this first you have to build the URL:
const presentation = SlidesApp.getActivePresentation();
const url = `https://docs.google.com/presentation/d/${presentation.getId()}/present`;
Then you might open a dialog or sidebar including client side code to open the URL as is done with any URL. For this you might use the function openURL from the Stephen's answer to Google Apps Script to open a URL:
/**
* Open a URL in a new tab.
*/
function openUrl( url ){
var html = HtmlService.createHtmlOutput('<html><script>'
+'window.close = function(){window.setTimeout(function(){google.script.host.close()},9)};'
+'var a = document.createElement("a"); a.href="'+url+'"; a.target="_blank";'
+'if(document.createEvent){'
+' var event=document.createEvent("MouseEvents");'
+' if(navigator.userAgent.toLowerCase().indexOf("firefox")>-1){window.document.body.append(a)}'
+' event.initEvent("click",true,true); a.dispatchEvent(event);'
+'}else{ a.click() }'
+'close();'
+'</script>'
// Offer URL as clickable link in case above code fails.
+'<body style="word-break:break-word;font-family:sans-serif;">Failed to open automatically. <a href="'+url+'" target="_blank" onclick="window.close()">Click here to proceed</a>.</body>'
+'<script>google.script.host.setHeight(40);google.script.host.setWidth(410)</script>'
+'</html>')
.setWidth( 90 ).setHeight( 1 );
SpreadsheetApp.getUi().showModalDialog( html, "Opening ..." );
}