You fix is pretty simple, you need to use ge-proton 9.23 as the default wine version and BNet will install just fine - some have also had success using proton hotfix (this does requires you to have installed steam and enabled it as a compatibilty tool as well as assign it a game to get it and other pieces downloaded).
This is straight from the lutris support discord server:
Battle.net update error on Lutris:
since latest update, Battle.Net has issues on Lutris and globally Wine.
But this issue are fixed in Proton-ge 9.23.
→ How to install Proton-ge 9.23 for Lutris
First be sure you use the latest Lutris version: 0.5.18
extract archive in this place:
Lutris native: /home/$USER/.local/share/lutris/runners/proton/
Lutris Flatpak: /home/$USER/.var/app/net.lutris.Lutris/data/lutris/runners/proton/
now restart Lutris and change Wine version associated to your Battle.net version to Proton-ge 9.23
Now Battle.Net should work as intended in Lutris.
Nota:
A) even if there is an error during Battle.Net install, the install stop towards 98%, you can ignore it, launch Battle.Net, everything works fine.
B) For Lutris native version, you will need to install umu-launcher to use Proton-ge and Steam Proton too. See their Github page: GitHub - Open-Wine-Components/umu-launcher (edited)