A solution for both HTTP and HTTPS contexts

The Clipboard API solution by @Joachim Lous was on the right track for me. However, this did not work on localhost, which used http not https. To get around this, I used a "copyToClipboard" function (adapted from this SO answer) which acts as an all-in-one wrapper function that takes into account a http context by using a clever "out of viewport hidden text area" trick.

Readable version:

javascript:

function copyToClipboard(textToCopy) {
    // navigator clipboard api needs a secure context (https)
    if (navigator.clipboard && window.isSecureContext) {
        // navigator clipboard api method
        return navigator.clipboard.writeText(textToCopy);
    } else {
        // use the 'out of viewport hidden text area' trick
        let textArea = document.createElement("textarea");
        textArea.value = textToCopy;
        // make the textarea out of viewport
        textArea.style.position = "fixed";
        textArea.style.left = "-999999px";
        textArea.style.top = "-999999px";
        document.body.appendChild(textArea);
        textArea.focus();
        textArea.select();
        return new Promise((res, rej) => {
            // here the magic happens
            document.execCommand('copy') ? res() : rej();
            textArea.remove();
        });
    }
};
var element = document.querySelector(".myClass");
var text = element.textContent || element.value;
copyToClipboard(text);

Simply replace ".myClass" with your selector.

Minified version for bookmarklet:

javascript:void function(){var a=document.querySelector(".myClass"),b=a.textContent||a.value;(function(a){if(navigator.clipboard&&window.isSecureContext)return navigator.clipboard.writeText(a);else{let b=document.createElement("textarea");return b.value=a,b.style.position="fixed",b.style.left="-999999px",b.style.top="-999999px",document.body.appendChild(b),b.focus(),b.select(),new Promise((a,c)=>{document.execCommand("copy")?a():c(),b.remove()})}})(b)}();