The store bought pets are a code, which can be "gifted" to another person. That's your workaround. Trust, that's the hang up
This. You can choose to "gift" the pet upon purchasing it from the Blizzard Pet Store, rather than redeeming it yourself.
I recommend extreme caution, though. Be very careful when choosing to "trade" it (gift it) to someone you don't know in exchange for a TCG pet. If the transaction goes awry, a Game Master probably won't be able to assist since it occurred mostly outside their jurisdiction.