Okay here's the skinny on battery backup for the purpose you want which is a very very occasional night of power failure.
Lithium ion batteries can do many more cycles before they die and lithium iron phosphate can do virtually infinite cycles. But having a battery that will last 3000 cycles is pointless for your usage because at one outage per eight years you'd be looking at several thousand years before the battery ran down and I suspect you would run down before that!
I'd recommend just getting a 35 amp hour AGM lead acid battery. If you're confident that you're just facing a single night of outage, you could get by with half the size of your battery. I built myself a battery backup system a few years ago because we were having a lot of outages due to fires. The cost was not very much. The battery on US amazon is about $75. A 20AH battery is about $45. There's a non OEM adapter for $31, I have one and it works great.
Do remember that typically you will get far less battery backup if you use your humidifier. But if you have enough battery you can use the humidifier at least for one night.
Here are some threads I posted. If you PM me I have a document with all the links to products.
viewtopic.php?f=1&t=180437&p=1366945#p1366945
viewtopic.php?f=1&t=180059&p=1363433#p1363433
The total parts list is: (prices may have changed)
Battery: Weize 12V 35AH Battery Rechargeable SLA Deep Cycle AGM $65
Battery Box: NOCO HM082BKS Group U1 Snap-Top Battery Box For Mobility And Lawn and Garden Batteries $11.36
Charger: NOCO GENIUS5, 5-Amp Fully-Automatic Smart Charger, 6V And 12V Battery Charger, Battery Maintainer, And Battery Desulfator With Temperature Compensation $55 on sale
Alligator to 12 volt accessory jack: NOCO GC017 12-Volt Adapter Plug Socket With Battery Clamps $9.94 This is fused.
Adapter for 12 volt to 24 volt for Resmed S9: KFD in Car Charger DC Adapter 3 Pin Jack for 90W Resmed S9 Series $33
Do you have a 12 volt or 24 volt machine?