The safes at the value resorts just have a key and are pretty small. I would think you would be better off using your ATM card and getting cash as you need it, or as the PP said getting like a Visa gift card you put the money on. You could do Disney dollars too if you plan to spend most of your money on-site. If you were in one of the resorts that has an actual safe you make the combination for, I would say you would be OK, but I would be a bit more nervous with the safes that just have the key.
Could you leave the cash in the bank and access w/an ATM card?
No, no, no. Not cash in the room safe. If anything happens to it you have no proof that it ever existed.
Take what you need for the day, and take the rest of it to the manager to be deposited in the hotel's main safe. They will give you a receipt for what you deposit with them. This way if any of it disappears, you have proof that it was there, and you can get reimbursed while they investigate the theft.
How about having the concierge apply some of it to your KTTW card? We do that with our travelers checks..it works really well. At the end of the week we just get cash back for what ever amt is left.