Hi Starlight Sparkle,
the newly added item via the “addNewTab” button always has a label of “New Tab” and it’s not possible to change that via API. However since the new item becomes the selected item by default you can accomplish the same result by binding to the change event and calling update method on it. Here’s how to do it:

document.querySelector('smart-tabs').addEventListener('change', function (event) {
        const newTabIndex = event.detail.index;
        //Get all tab items inside the element
        const allTabItems = this.querySelectorAll('smart-tab-item');
        //Update the label
        if (allTabItems[newTabIndex].label === 'New Tab') {
            this.update(newTabIndex, 'New Custom Label');

Best Regards,
Smart HTML Elements Team