Even after a cold crash there is plenty of yeast left to ferment whatever new sugar is added. If you add sugar and then bottle with whatever sugar you'd normally use for carbonation, you'll get bottle bombs. If you keep things cold and keg (force carbonate) you'd probably be OK with the sugar in there.
But then the more important question.....why would you want to sweeten beer in the first place? Unlike cider and mead, most beers taste awful with any residual simple sugars - they are just too sweet. Sugar sweetness tastes quite different to residual malt sugar (eg. dextrin) body (which is nowhere near as sweet). The only beer style that I can think of that uses some added sweetness is a milk stout - it uses lactose which is not fermentable by brewers yeast. Again, lactose doesn't taste anywhere near as sweet as sucrose/glucose/fructose. Before you try this with a batch of beer, try adding some dissolved sugar to a glass of beer - it does not taste good.