Isn’t the mouse mount for just playing Merc bounties, similar to the flying cardback mount or the horse from HoTS? I think Out of Cards got something mixed up in their post
(https://outof.cards/hearthstone/3574-hearthstones-sarge-enters-world-of-warcraft-as-a-rideable-mount-earned-through-a-6-month-subscription)
You could look into WoW’s Mount tab, if the mouse is already displayed you should see how to get it.