Just to understand, are you saying your CPAP runs off 110v AND 12v or did you mean 110v OR 12v?
If your 110v goes out, then so will the converter so it's better to run it from the battery with a fuse. Why don't you just skip the 12 outlet and install an "Inverter". Find out the wattage capacity for your CPAP then install the inverter which automatically converts the 12v battery to 110v for your CPAP. In other forums I belong to this is a very successful option. The reason is that the CPAP will always have power (from the batteries) and you don't have to worry about 110v not being available. When 110v is available then the converter helps keep the batteries charged anyway. Your only concern is keeping the batteries charged when not connected to shore power.
The CPAP runs plugs into the inverter just like if would from the 110v outlet. The biggest problem you'd have to consider is where to mount the inverter to get to the outlets and how to get the 12v inverter wires back to the battery. But you'd have the same problem running the wires from the 12v outlet (lighter).
Inverters are sold in many wattages. As low at 100w, to 3,000 and even higher. I had a 1000W inverter in a prior trailer and that was plenty. It was wired to the battery (with a fuse) and gave me 110v power when not connected to shore power. I used it for air compressors, and power drills when I stored the trailer elsewhere and didn't have 110v close by. The inverter should be installed by someone who is well versed in electronics. I had it done by someone since it need an isolator and I don't know how to do that. An isolator helps protect the trailer when using multiple power sources (truck supply, shore power, solar panels or whatever you happen to have).
If might cost more overall, but it will be money well spent, over the lighter solution.