Purchase a cpap that is 12 Volt capable . That is , works directly from a battery without an inverter. Then purchase the power cord and connect it to a 12 volt deep cycle battery . Most use a M connector available at radio shack. I have used mine for over 2 weeks on a single charge.. You can also connect it permenantly to your battery . You wil need a charger to do this. The charger will keep it charged and you will not need to do anything special should the power fail. Just keep sleeping.
Total cost is about $125