Larry,
Like some of the others have done, I too have added a 12 VDC compressor and tank to my truck. I carry 50' of air hose so I can reach all the tires on the RV from the truck. I think that setup was maybe
$400.
For you, needing the Bi-Pap when boondocking, I too would lean more towards the inverter system. That system would be an inverter sized to at least cover the draw of your compressor for a decent amount of time, paired with a battery bank to sustain DV power to the inverter as needed.
Many on here may now know the current requirements of your bi-pap device. Take a look at it, where the power cord connects to it and look for the power requirements there. Look for the voltage, amperage and/or wattage. Or, reply back here with the brand and model and we'll figure it out for you. My sense it that it is a pretty small requirement. And nothing like what your compressor will need. That thing will likely be a power hog. Here again, look at your compressor for the requirements or provide brand/model here and we'll look it up.
If your compressor is similar to Forrest's, you may need an inverter no larger than he has. I think his is 2,000 watts.
For batteries, as others have suggested, at least 2 Trojan T105 Golf Cart batteries are a good choice. But there are many other decent choices too. Sams/Costco carries a good supply of alternate brands of deep cycle batteries as well. And sometimes, Costco carries the 2,000 watt Xantrex inverter. It's the cheaper, MSW (modified sine wave) model but your bi-pap and compressor may work fine with it.
Find yourself an installer for this sort of work and tell them what your requirements are and what suggestions you've been given here are and go from there.
Good luck.
Jim