I have no particular expertise on this, but my reaction is that you should not take expenses from winnings, because it muddies the water a little, and leaves you open to (unfounded) accusations of benefiting from the charities money. Much better to be able to say "All winnings to charity".
I would be very optimistic that you could raise donations to cover the expenses (as others have suggested) - you just need to work out the cleanest way to collect the money, and make sure that route has the expenses point clearly stated. If there is nothing in APAT rules that prevents you publishing occasional requests, I think APAT members will cover it easily.
I would certainly not hesitate to put a donation to expenses, if you give me an easy way (eg Stars name, bank details). I have a lot of admiration for people who put themselves out for this kind of cause.