I am not convinced that o2 use is optional in a G-reg on a private flight -
ANO amendment
A while ago I wrote
this up and (if you don't want to read it all) if you scroll to the very end you can see what I currently fly with i.e. the MH composite cylinder, MH 1st stage reg, O2D2 electronic demand reg, and the plain cannulas. This is simply the best kit.
For low level use i.e. below say FL150, you would be fine without the O2D2 and just using the oxymiser/oxysaver cannulas, but you will get through a lot more gas (
comparison) which is OK if you have a big cylinder and have a handy refill facility back home (some have, some haven't).
Whether
you will be OK is impossible to say. I once flew with an instructor, who refused o2 saying he doesn't need it. At FL120 he could not read the altimeter... which looking at his general fitness was not suprising. And he wasn't even a smoker. On o2, he was fine within seconds.
I use o2 anytime at FL100 or above, and one arrives fresh instead of shagged

This is despite having once participated in a flight test to FL120 with a group of pilots and I was way better than any of the others in the blood o2 and heart rate figures.
The other thing is that if you fly so close to the mountain tops, you are vulnerable to up/downdraughts caused by wind flowing over them. If you are say 1000ft above, even 10kt of wind could easily kill your ability to hold altitude if you are near your operating ceiling there. You need to either pick a very calm day, or fly a fair bit higher (and then you really must have o2). In my
trip reports you can see how I go about this, but then I can go to FL200 which is very handy.
Sticking to a preplanned MSA in mountains is important (if one is flying that close) because of the risk of false horizons.
IMC over the Alps is tricky because the OAT is likely to be below 0C even in the summer. I have done many crossings (straight over the top; I don't do "mountain flying") and have never seen a positive OAT. So there is a risk of structural icing, and you have cut off the best escape route which is a descent into warmer air but without going below the MSA.
a CAA pilot is entitled to fly an N reg aircraft in UK airspace - presumably said pilot is not required to comply with the FARs since he does not hold a FAA ticket? I wonder?
He is required to comply with FAA regs because he is still flying an
N-reg but is merely using non-US pilot papers, permitted in accordance with FAR 61.3.