Successfully ran the Blizzard App on a 32-Bit and 64-Bit Wine, while version was set to "Windows 7". Used this Wine-Staging build on a openSUSE 42.2 (x86_64) system:
download.opensuse.org/repositories/Emulators:/Wine/openSUSE_Leap_42.2/x86_64/wine-staging-2.8-665.1.x86_64.rpm
download.opensuse.org/repositories/Emulators:/Wine/openSUSE_Leap_42.2/x86_64/wine-staging-32bit-2.8-665.1.x86_64.rpm

Only problem: There where two "E-Mail" and two "Password" fields in the login window and I had to enter my login data in the lower ones.

Have you successfully run any game though?

Yes, everything works fine! (campaign, multiplayer)

Confirmed working on funtoo with wine 2.3 after enabling the staging USE flag. Only tested 32-bit profile.

I guess it's running with Wine-Staging because of this patches.
(I wasn't able to do exact tests until now)

dev.wine-staging.com/patches/99/

dev.wine-staging.com/patches/103/

dev.wine-staging.com/patches/104/

dev.wine-staging.com/patches/107/

With wine staging 2.9 (CSMT on, of course) and version set to "Windows 7" on openSUSE 42.2 (x86_64) runs OK.

Only 2 errors.

1 - Two e-mail and passwords fields. Only type in the lower ones and uncheking (every day) the "no close session" option.

2 - Don't feed anything in Games panels (other tabs as Shop and News works OK). I can't see players on games nor Frame of news or eventually frames of warnings of problems or semanal maintenances. In SC and OW not appear the buttons Try free.

The point what bothers me the most is the second Point. The first point is annoying but passable.

This is for correct myself. In tabs Shop and News I only see text. 0 images.