I still have power and the display shows 11.4 v
This is a symptom rather than a cause. Still the same symptom with two new batteries so unlikely to be due to a dud cell.
I might have missed it, but has anyone suggested part of the reason might be that the batteries are not being charged up properly in the first place. A fixed voltage converter does a good job of keeping fully-charged batteries fully charged, but a fairly poor job of recharging flat batteries unless you have a couple of days to spare. Even something like a 30 amp three-stage charger would take about 12 hours to fully recharge dead flat batteries of around 240 Ah capacity.
Plenty of people run CPAPs with fairly rudimentary 12V systems, but only if they don't use the humidifier/heater. With that in the circuit, you would need to double or even triple the size of your battery storage, especially if running off an inverter
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RVs. USA - Airstream Cutter; in Australia - MC8 40' DIY Coach conversion & OKA 4x4 MH; in Germany - Hobby Class C; in S America - F350 with 2500 10.6 Bigfoot camper