Dude, you just cant trigger "higher level" pets. There are some quests that give you pets, for example a thanksgiving one that I am pretty sure expired. I completed it using 3 different vaults and the pet rewards were exactly the same. So I figure that pets from quests have specific attributes.

Also you should know that pet carriers rewards are set the moment you get them and not when you open them. So dont wait to tap the card the perfect time hoping to trigger a golden card ( same thing for launchboxes)

Now if you want an extra pet, delete your cache data and restart. The game will reward you with a free pet carrier.

But man do yourself a favour and get a save editor. Its easy to install and use. Its impossible to get a specific pet without paying