How do I install StarCraft 2: Legacy of the Void through the Blizzard client?

According to my battle.net account I own StarCraft II: Legacy of the Void. There's a download button for me to get my copy if I want.

However, I'd prefer to download and play it through the Blizzard desktop app. When I install that and go to the StarCraft II section it says it's the "starter edition" and there's a "Try for Free" button.

I cannot see any way to use the client to play Legacy of the Void. There's nothing in the drop-down menu for regions (which I understand is where it was during the beta). There doesn't seem to be any more recent advice for this that I can find.

Is it possible? If so, how?

I contacted Blizzard support, as suggested. This was their reply.

I apologise for the confusion here, you do in fact own StarCraft 2, but it hasn't yet triggered on the app - to get it to update, just download and play the trial and when you login select a campaign and then log out, it will then correctly show that you own it :)

Having now done this, I can confirm that it works. Seems a bit of a UI oversight but there you go.