I have a short position that I have a significant profit on. Is it possible to donate this to charity the same way I might donate "appreciated stock", so as to avoid the short-term capital gains?
No. There is no asset associated with your short position, so there's nothing to gift. The short position in the stock is purely a liability. When you note that you have a profit in the position, what you mean is that the cash you made when you shorted the stock is more than enough to cover the short position. The only asset in this picture, then, is the cash you made when you entered the short position.
With a short position you make your money (profit) when you buy the stocks back to close the position at a lower price than what you bought them at. As short selling is classed as speculation and not investing and you at no time own any actual assets, you cannot donate any short possition to charity.
If you did want to avoid paying tax on the profits you could donate the proceeds of the profits after closing the position and thus get a tax deduction equal to the profits you made.
But that raises a new and more important question, why are you trading in the first place if you are afraid to make profits in case you have to pay tax on those profits?