You can simply buy a reasonably hefty size diode though which you connect the second battery to the FJR charging system, this way the second battery is always being charged when the bike is running but you never need to worry about draining the FJR battery.
The problem with doing this is the aux battery will never be fully charged, it will be one diode drop lower than the main battery.
Best idea would be to find a CPAP machine that is designed to be low power and run off its own rechargable battery, which charges from a 12V cigarette lighter outlet. Charge the thing while you're riding, and unless you're camping where there's no electrical outlets, run it off 120V AC whenever possible.
The
Everest machine previously linked to looks exactly like what the OP needs.
I've emailed the manufacturer to ask how much current the adapter draws & will post the answer here.
Their answer:
12VDC supply current drain with Mobile Power Adapter (assuming 90% efficiency):
0.63Amps @ 4cmH2O.
1.43Amps @ 12cmH2O.
2.34Amps @20cmH2O.
Note: Normally, the AEIOMed battery is charged via the car battery and Mobile Power Adapter during the day and the CPAP is powered by the AEIOMed battery at night. In that case, the drain on the car battery is in the 8 to 10 AmpHr range per day.
I then asked what the current drain would be when charging the battery, rather than running the unit off the cigarette lighter adapter, and I told them that the bike's 12V outlet was limited to 3A:
The continuous current would be 2.7 amps however there is an initial surge of 6 amps which in your case would present a challenge.