Carbon fiber would save weight but it would be such a small amount of weight you wouldn't notice. In addition, carbon fiber fails explosively...imagine glass shards. Under the rare circumstance you get in an accident or run over a giant rock and it happens to jam up against it until it fails, all that shrapnel will fly into your engine bay and you'll never be able to clean it all out, not to mention what will jam into any hoses on the underside of your engine.
The car doesn't overheat without the undertray, but it is harder work for the car to stay cool. Imagine running a mile in 10:00...quite possible for most people. Now imagine running a mile with a backpack filled with sand on your back in under 10:00. Doable, but it'll work you a lot harder.
Washers will help alleviate the problem, yes. The more you can distribute the stress the better. The undertray seems to fail most often at those bolt holes because of the small space between the holes and the outside edge of the tray. If you buy a new one don't expect it to break overnight, it'll last more than a few years under regular driving conditions