Recently, there were was an upgrade to the Battle.net client that caused it to break on Linux desktop / Steam Deck but it seems the issue is thankfully solved now.
The thing is, Blizzard don't support Linux as a platform at all for the client or their games. However, they do have a bit of history of now and then fixing up major issues preventing people playing with the Wine compatibility layer. Seems that level of support is still ongoing after the latest incident.
Around 9 days ago, it seems an update just broke it entirely and the app wouldn't install and update properly resulting in some sort of loop. While it's a shame no testing is done for Linux and Steam Deck before updates are rolled out, at least we're not stuck forever as they've now solved it:
I do apologies we didn’t post here sooner but the development teams was aware that a recent change did cause havoc with Wine and the software used for Steam Deck. This wasn’t intended.
A fix was deployed and it appears it’s working. If you are still having issues please try a reinstall of the app so it fully updates to the 8020 build.
Check on it today and it seems all good on my end now too.
Good news, since Overwatch 2 releases next month and I'm sure plenty will want to play it.