Pulse vs. continuous flow oxygen converter

I recently started using oxygen at night.  Because I travel I wanted a portable unit.  The provider gave me a Sequel Eclipse which, in my opinion, is too big for use on an airplane.  I now have two units, a large continuous flow unit for use at night at home and a FfeeStyle by AirSep pulse unit to use when traveling.  I was told that I should not use the pulse unit at night because you breathe through your mouth at night and, therefore, don't get enough oxygen with a pulse unit.  (Apparently it's ok to use it at night on the road though.)

My question is: if you only get oxygen from both units when you breathe in through your nose, what's the difference? It seems to me that you'd get just as much oxygen from the pulse unit if the continuous unit only supplied oxygen when you breathe in through your nose.

I really got suspicious when I found out the provider had been billing Medicare for two units - a permanent and a portable model - because Medicare classifies the Sequel as both.  They were getting paid twice for one unit.  Now the are being paid for two units but I wonder if I couldn't use the pulse model for both day and night use and save the taxpayers a few bucks.