Ah launchers, don't you just love them? A reoccuring nuisance for Steam Deck and Desktop Linux gaming is that they can and will break and the latest to break (again) is Battle.net. Here's how you can fix it for now.
Firstly, for people just using Proton on Steam Deck and Desktop Linux direct through Steam you can use the bleeding-edge Beta of Proton Experimental. As posted on X (Twitter) by Valve developer Pierre-Loup Griffais:
How to do that? Find Proton Experimental in your Steam Library and go into its Properties -> Betas -> and select bleeding-edge from the dropdown box.
Another solution that you can try is to use this as a launch option on any Proton version in Steam:
For people using some form of Wine / Proton in Lutris you can add WINE_SIMULATE_WRITECOPY with a value of 1 as an environment variable in the system options inside a Wine runner you're using.
It can be frustrating when this happens, and it's certainly far from the first time Battle.net has broken along with EA, Ubisoft and other extra launchers. Thankfully this time it's an easy enough sort until it's fixed properly inside Wine and Proton directly.
Update - 17:17 UTC: If you use Wine-GE, the latest Wine-GE-Proton8-25 release has a fix for it.