Can I use JavaScript fetch() to insert an inline svg in the DOM?

You can. I think the following should work:

fetch('bar.svg')
    .then(r => r.text())
    .then(text => {
        el.innerHTML = text;
    })
    .catch(console.error.bind(console));

The second then is for resolving the .text(), which "takes a Response stream and reads it to completion" (MDN) (MDN Eng).