There are a few possible cases here:
Case a) If the amperage you're quoting is at 120V, then your CPAP machine draws 120V * 1A~1.5A = 120~180 watts.
Case b) If the amperage is at 12V, then your CPAP machine draws 12V * 1A~1.5A = 12~18 watts. That's a pretty significant difference.
A cigarette lighter plug-in style inverter will be limited to drawing about 15 amps at 12 volts, which is 180W. With case a), you're right at the limit for using that type of connection. In either case a hair dryer is simply not an option unless the inverter is hard-wired.
As mentioned, your CPAP will very likely require a pure sine-wave inverter. Something like this would be a good choice:
https://www.amazon.com/Samlex-Solar-PST-300-12-Pure-Inverter/dp/B00H8N97E2
For case a), running a 120~180 watt appliance for 8 hours overnight would require between 0.96 and 1.44 kWh. That's a fair amount of power. A pair of T-105s taken from 100% to 50% SoC over ten hours is good for about 187 Ah, or 1.12 kWh.
While you may be able to run a hair dryer for brief use from those batteries with a large enough inverter, most people here would recommend against it as the inverter and wiring would be a large investment (hundreds of dollars) for a single purpose use.
I don't have a CPAP but from reading about them, the humidifier is the largest current draw, and it may not be running continously so the numbers on the nameplate may not be reflective of actual power consumption.