Company of Heroes 2 Beta - I want full version! What should I do?

Go here: https://store.steampowered.com/account/
And try clicking "remove" next to the game's title.

If it's not there, contact Steam support and request the removal of the game.

Btw. if you consider buying it, check out the Humble Weekly for 5$
https://www.humblebundle.com/weekly

The stand alone multiplayer will add CoH2 to your library but in-game if you try to play the campaign you will see a "buy" button in case you want to unlock it.
Don't worry about the "beta" version apperaing also on your library, you can hide it if you want.

You're able to activate keys for CoH2 just fine, if you're unable to buy from the steam store (which i'm guessing isn't the case atm) you can buy it as a gift, then redeem that gift to circumvent this minor bug.

The base game itself is considered DLC, this is because even if you only have one of the stand alone DLC's you'll still need all the assets from other dlc's/the base game to be able to play skirmishes/MP battles, so like ignacio said you'll have the full game installed but the campaign and other factions will have a "buy" button instead of a "play" button.