Since your must use the ResMed 12/24 converter, why not go with a 12 V AGM battery in the 35 Amp-hour range.
The Deltran Battery Tenders are kind to Sealed AGM batteries.
There are folks who keep an AGM battery on a Battery Tender; and hook up manually during power outage.
There is also the crowd that wants to take their system camping.
It sounds like you'd like to rig up sort of a UPS system where the Cpap and converter run off 12 VDC. Pulling from a "Battery Charger" when the mains are live; and from the battery when the mains are down -- with automatic switch-over. For that, you will need a power supply like the Astron SS-25M, 12 V supply, a method of switching like the West Mountain Powergate, a Battery, and your converter.
You want to Float Charge for the above method. Not rely on a Battery Tender to power your CPAP when the mains are live.
You should talk with user OMyMyOHellYes. He used such a solution. You have to be electrically very handy. You'll need an accurate DVM to set the power supply Voltage. You'll need the tools to make up cabling using Anderson Power poles.
If you just want to go simple -- then check out this guys setup:
http://www.apneaboard.com/forums/Thread-...#pid106570
His is for off main use; and does not have automatic switch over.