I have both a ResMed AirSense 10 and a Medistrom external battery; I use the battery at home (as a pass-through to deal with any power loss) as well as on trips.

I've never had anyone question it as an "extra" carry on. The travel case is fairly common, so I've put both one of my DM cowbell metal tags and a distinctive luggage tag into it to avoid confusion when getting it out of overhead bins.

I've used it overnight on both DL and VS metal with the battery, not bothering to plug the battery in to charge in either case. VS did ask me to confirm that I was not using the aircraft power. The battery is more than enough for SEA-LHR with the heated hose and humidifier turned off; IIRC it still showed half full at the end.

The ResMed manual explicitly says that you should not put water in the chamber when using it on an airplane to avoid turbulence spilling it into the machine (or presumably the hose).