Unfortunately, it looks like the spec was built to favor consistent native behavior over in-browser customization.
We did a bunch of research and summarized our findings here:
https://www.inkling.com/engineering/native-html5-drag-drop/
The long and short, though, is that using a helper (like jQuery UI) or rolling your own is generally preferable if you need to do anything sophisticated.
Still, if you'd really like to use the native behavior and setDragImage() isn't adequate, there are a few alternatives.
One particularly extravagant approach could involve hijacking the renderer to create a screenshot (!) of the element as you‘d like it to be styled, and then inserting it via a data URI.
See: http://cburgmer.github.io/rasterizeHTML.js/jsconfeu2013/#/step-4
A saner approach would be to simply absolutely position a ghost element to follow the mouse. But that takes us back to writing code that re-implements core parts of drag behavior outside of the API.