Hello vermu,
Alpha channel is always exported when EXR is used.
Photoshop usually keeps the alpha into “Channels” tab - not in the “Layers” tab.
If you switch to “Channels” tab you should see - Red/Green/Blue and Alpha channels.
Another option to get rid of HDR background is to turn on “invisible” check-box into Dome-Light parameters - this way dome light will illuminate your scene and it will be invisible in the background.