I know this is old but I am flirting with going to Philmont with my scouts and I too have Sleep Apnea.
I have been using a CPAP at camp now since 2016. At first I brought my full size CPAP and a long extension cord. Then I got more portable by getting a deep cell battery (not as big as a car battery) and a DC converter. This was a pain in the ass and the battery keep loosing its charge cause I was only using it once a month. I then tried the freedom batteries and they worked ok but, where very expensive, heavy and did not last more than 2 nights. Still I had this large CPAP... I then decided to look at travel CPAPs which were really expensive.
I found an article about this guy who traveled to Mount Kilimanjaro in Africa. He laid out what he brought and for the last 3 years I now have a similar CPAP camping kit:
Z2 Standard Travel CPAP Machine
TalentCell Rechargeable 12V/11000mAh DC Output Li-ion Battery Pack (lasts 2 days CPAP set at 13lbs)
TalentCell 12V Lithium ion Battery PB120B1, Rechargeable 12800mAh Li-ion Battery Pack (lasts 3 days CPAP set at 13lbs)
Fireproof Bag LiPo Battery Bag
SMAKN DC-DC Power Converter 12V to 15V 5A Boost DC Power Module
Travel Bag for Small CPAP Machine
Mask, short tube and spare filters
Heat Moisture Exchange Unit for CPAP Machines
My CPAP camping kit weights in at 5.31 lbs. The 11000mAh battery lasts 3 nights and the 38400mAh lasts 2 nights. I have been thinking of getting a solar panel to charge on hikes. I am a big guy at 6'5" (currently 330 lbs need to get down to 265 lbs for Philmont) so an extra 5.5 lbs of weight does not kill me; sleeping with out a CPAP probably would kill me at some point.
To go 14 days hiking I would either have to have a solar charger that could charge a battery in a day or bring a lot of batteries or a combo of both.
I have well over 150 nights camping with a CPAP with temps at night 90+ degrees to 20 degrees. Only had two issues: battery management (not charging day before, forgetting a power cord, leaving the battery on all day, etc) and an old ResMed converter caught fire during my Ordeal. I have not gone back packing with it yet but plan to try it out well before I get on a plane.
Goals:
Loose approx 70 lbs
build my endurance for the trip
Several weekend back pack trips at some elevation (that means driving north as I live in Florida)
Work out battery management with solar charging.