Welcome to the forum, adamfrey!
I assume you have a Travato 59G and have a 1000W inverter that come standard with it. If you don't please start a new post and include the model you have and we can provide a better answer.
The 350W oxygen concentrator should be able to be run from the inverter indefinitely while driving. 350W will draw about 35A from your 12V alternator, through the separator relay to the house battery system, then out the inverter (where it gets converted to 120VAC) to the outlet.
Note that when the vehicle is off, you'll deplete your house battery fairly quickly, as that 35A will now have to come out of your house battery. The two group 31 batteries that are standard with the G are typically 100AH each and it's best practice to discharge no more than 50%, so you have 100AH to work with.
The 35A load by itself will discharge the battery to 50% in a little less than 3 hours. This is on top of other loads like lights, heater, electronic devices, etc, so the time will be less. IMHO hooking up to shore power or running the generator would be advisable when the engine is off for more than a couple of hours with the concentrator on.
Good Luck, and I'm sure you'll have some great experiences and lasting memories during your travels with your Mom!
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Stewart, Brenda and kids
2006 Roadtrek 210 Versatile, 400W Solar, 320AH LiFePO4. Suspension mods: Front: Moog 81004 coil springs. Rear: SuperSteer Track Bar, AirLift airbags with integrated bump stop, 2" lift blocks, removed overload leafs. All around: Bilsteins, 265-75R16 K02's
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