I have flown - repeatedly - over several mountain ranges, not only the Alps. Leaving the O2 issue aside, trying to cross ANY mountain range with only 1000ft to spare borders on the insane, especially in an (at that altitude) totally underpowered a/c.
Do *not* assume that because it's nice and calm where you start your trip and the fcst wx is good that it will be like that at altitude and/or remain throughout your trip. Wx in the Alps especially can change at no notice at all and can easily catch you out. Always have a Plan B and enough air between you and the cumulugranitus.
Btw, you say you are in southern Germany. IIRC getting to FL120/150 is not an issue in Austria, if you want to give it a go. Then again, you'll already be half way across the Alps by then
