Reinstall OS X hangs on AppleID login

I bought a new SSD for my 2011 Macbook Pro. I booted with CMD-R (and Shift-Alt-CMD-R) and I could erase/partition the new drive. However when I select the drive during the installation process it asks me to sign in with my AppleID to download from the App Store. This login is systematically refused.

I've reset my password (although I could login with the old credentials on https://appleid.apple.com/ )

and I've tried creating a new AppleID. Both ID's login fine on https://appleid.apple.com/ but both ID's are refused in the installation process.

Error messages vary between 'Your Apple ID or password is incorrect' or I get a button for 'Forgot password' which does nothing when you click on it.

I do get 2FA notifications via SMS or other Apple devices, but I cannot enter the code during the installation process.

Any suggestions on how to proceed?

(ps, Downloading the image from Mac OS X Lion Installer and install from USB is impossible. Both USB ports are broken :-( )