There is no API in Vaadin 24 yet for setting overlay classname of the tooltip. You can use the following JavaScript utility function with Vaadin 24.2.0 and nwer.

TooltipUtils.java

public class TooltipUtils {

    public static void setTooltipClassName(Component comp, String className) {
        comp.getElement().executeJs(
        """
            if ($0.getElementsByTagName('vaadin-tooltip').length == 1) {
                $0.getElementsByTagName('vaadin-tooltip')[0]._overlayElement.setAttribute('class',$1);
            } else {
                const tooltips = document.getElementsByTagName('vaadin-tooltip');
                for (let i=0; i<tooltips.length; i++ ) {
                    const tooltip = tooltips[i];
                    if (tooltip.children[0].id === $0.getAttribute('aria-describedBy')) {
                        tooltip._overlayElement.setAttribute('class',$1)
                    }
                }
            }
         """, comp.getElement(), className);
    }
}