I always suggest bidding on a 3* property in the MCO area through Priceline...usually a bid in the mid $40's can get a room at either the Hyatt inside the airport, the Holiday Inn Select, the Marriott, the Renaissance or the Sheraton Suites....all have complimentary shuttle service 24/7, nice pools and are very nice properties (none include complimentary breakfast, although for what you save you can more than buy breakfast as all have nice places to eat).
Sometimes you get lucky and get the Hyatt inside the airport although not too often....although I have decided the rooms are substantially larger at the other places and with their excellent shuttle services they are just as easy to use.
As far as messing up the DME service....you would need to claim your own bags on arrival (do NOT use the DME tags they send you) and take them with you to the hotel over night. The next AM you could return to the airport via the hotel's complimentary shuttle service with your luggage and go to the DME check in desk with your bags. (you would need to have booked your DME for the AM you plan to travel to Disney, NOT for the evening you arrived. Since they ask for flight information you will probably need to find a flight that arrives sometime around the time you want to take the DME to WDW and provide those flight #'s to minimize confusion)...but since you will collect your bags and take them to DME check in yourself it should not be a problem.
