Interestingly enough, big cats work really well here. The Leo personality demands to be seen, while the INTJ personality is territorial and likes to be in control.
As such, you want an animal that’s visible, territorial, and dominant. As a Slytherin, there’s also ambition and cunning to think about, so a hunter or predator might be appropriate. And, of course, there’s resourcefulness - you’re going to want to camouflage yourself sometimes.
Of course, a lion would just be too obvious.
A leopard or panther would actually be my best guess. While its spots make it visible to us most of the time, in their native environments, they blend in much more easily - it’s good to be seen, but not always. Being seen often also means showing your hand and making yourself vulnerable. Leopards and panthers hunt alone at night and sleep during the day, which makes them less of an easy target for humans who want to kill them. As a result, they’re the most numerous species of big cat in the world, numbering at between 175,000 and 250,000.
And, of course, they possess the brutal elegance demanded by our house. Achingly beautiful to watch, terrifyingly effective at killing their prey.
Other possible Patroni might include the praying mantis. One of the largest insect species in the world, they won’t go unnoticed unless they’re in their natural environment. Their long green bodies let them hide in the bushes where they live, until an unsuspecting critter (which can and does include lizards, birds, frog, fish, and mice) comes by. The silent hunter will then snap up their prospective prey in their spiked, jointed forelegs and chow down.
Like any other animal described here, mantids also have the intention of being seen - and feared - by potential attackers. They throw their limbs up, unfold their wings, and hiss to create a threat display that makes them seem much more formidable. Some mantids even have brightly colored wings to enhance this effect.
Of course, we snakes do have one of our own that does this - look no further than the king cobra. Largest of the venomous snakes, this sly serpent’s scale patterns let it conceal itself in the bark of trees or the dried, dead leaves on the rainforest floor. It hunts just as quietly and carefully as any other snake, and strikes with blinding speed and deadly venom. Also incredibly territorial, it will rear up, flare out its trademark hood, hiss, and finally strike an invader with a painful set of fangs full of neurotoxins.
As an INTJ Leo, you were practically built for this house. In your comfort zone, you’re ruthlessly efficient at what you do. Faced with adversity, you dream big and fight hard for what you want out of life. Just winning isn’t enough - you have to win conspicuously, so people can see for themselves just how much of a badass you are. You embody the best of Slytherin’s virtues to a level that’s nothing less than appropriately legendary. The finesse and ferocity of the leopard, mantis and snake are so renowned that humans have created martial arts styles mimicking them. True power and the ambition for it ultimately mean the ability and will to survive, and you practically have that built into your mental genes. When people talk about nature being about the survival of the fittest, they’re talking about you.