I did cross the Alps with Racebike and with MTB. Highest altitude was with MTB, about 3000m.
Probably it depends a bit on your overall fitness and how sensitive your body is concerning altitude.
But if your legs feel like "lead" it might just be because you climb several 1000m without any break.
For instance it takes several hours to climb the "Stilfser Joch" offroad with a MTB. After that you are tired anyway, no matter how high it is.
The feeling of climbing for several hours probably makes you thinking you are lacking oxygen, but you are just getting tired that's it.
Science has proven that noticeable performance impact won't occure at altitude's less than 2400m.
From what I have experienced I would totally agree with that.
Remember to bring some armwarmers, leg warmers a west and gloves. Temperature will decrease by 4°C for every 1000m.
It can rain at any time even if the weather report tells you it shouldn't rain, so bring a light rain jacket as well.
In the Alps, weather can have a far bigger impact on your bike ride than altitude. You easily can have snow and hail in mid summer. The most predictable weather will be from late August to end of September.