Who said you could not have more than 6 pets in one room.

Ensure a triple room has at least 3 dwellers that have pets assigned to them.
You could add another 3 dwellers, but that just complicate this process.
Place another dwellers with pets nearby.
Once the vault is UNDER ATTACK move 3 dwellers to a room so that you end up with 6 dwellers and 6 pets.
Even better if all 6 pets have + damage properties.
Once the incident is over, they will auto return to their original rooms - just like normal.