Manuel 🇫🇷 Hey Manuel, first a quick info that Together has a built in translator, Sander actually replied in English :)
There is offline PIN and offline payments, but offline PIN also works with online payments and it's veeery common in France to use offline PIN, even if the payment itself is online. Dual PIN can only work with Online PIN terminals, as you can't store 2 PINs on one chip and that's why it fails for you.
When you did a cash withdrawal successfully offline payments (and offline PIN!) should work. Do you see any event in the bunq app showing up when you try to pay? Any specific error at the gas station? I personally could already use my Travel Card on a plane completely offline (the payment is still not showing up), and other people could also use it successfully already.
It might be an issue with the specific gas station terminal.. when you say "offline never works either" - how often and at how many different places did you try it? Even in France offline payments shouldn't be that common and gas stations aren't always offline either, especially in Europe, that's more of an issue outside of Europe.