So, the story is about the heroine of the book, but the whole thing is narrated by her pet. Initially the plot is light and easy, but later into the story, it gets a little serious. There's stuff like death and blood bath. Now here, in the few scenes I'm going to describe the 'bloody incident', I want to add MY two (philosophical) cents on the tragedy, with stuff that includes death, beauty and darkness, through the pet's narration. How can I do that without making the reader think - 'Ok, now this is unreal. How can an animal THINK such philosophical stuff?' Also is it safe to take this approach at all? Should I simply make the narration a series of events told from a pet's POV with no philosophical stuff at all?
Having an animal philosophize is not a problem. What you do need to be mindful is not the how but the why. It can't be your two cents—it must be your character's. Why is your character thinking such stuff? If you can't come up with a reason why, they shouldn't be thinking about it.
(Unlike your typical third person narrative, your narrator is likely a character. Even if the pet isn't in the story, they will still be present in the narrative by bringing a perspective only an animal can have, in addition to having an emotional stake in the story, since it is about the pet's owner.)
A good example here is The Art of Racing in the Rain because the narrator/main character is Enzo, a dog who is described as being "a philosopher with a nearly human soul". A lot of Enzo's philosophy is based off race car driving. Why would a dog know anything about that? Because his owner is a race car driver who describes to Enzo how it's done. Enzo is also able to speak intelligently about a wide range of other human topics because he has watched so much TV. Additionally, Enzo's actions are shaped by his perspective as a dog. For example, he is fiercely loyal and protective of the people he cares about.
Provided you believe your narrator can philosophize, the reader will accept it.
Many stories are told about animals and not all of them are light and suitable for children.
You have an opportunity here to provide commentary on your MC from the point of view of someone who loves her but, depending on species, might think it remarkable that she has managed to figure out how to turn a door knob. I had a cat, who, when he wanted to go outside, would reach up and almost touch the doorknob. Here, turn that this way.... No, rotate it like this - a few more degrees and the bolt will release. Honestly, bigger animals are just so dumb. Poor humans.
A friend of mine who I was helpng with her horse once said she wished he could talk. I asked her if she really wanted to hear what he might have to say, would she expect sweet remarks or something a bit more caustic about the injustice of the relationship?
Intelligence comes in many forms. Humans will often discount what a mere animal will do. Surely, no mere animal has problem solving skills, no animal could actually think. There were some trappers in Scandinavia who accused each other of theft. Traps were laid out and then tripped, but were found empty. Cameras were set up to catch the thief in the act, only the culprit was avian. A raven had determined the purpose of the traps and would fly the trap line. Anytime it found something snared that it could eat, it did.
I had a flock of bantams when I was a kid - odd pets, but very instructive. I observed behaviour that can best be described as honourable. I saw one rooster being attacked by two others, so the king of the flock stepped in (only after the second assailant joined in) and fought them both. Two against one? Not happening in my flock! Chickens have a complete language - I forget which study indicated it - but it makes sense.
Just find your narrator’s voice and run with it. You are not inventing the wheel - animals have narrated it is normally stories about other animals. You will just have the narrator cat/dog think and comment on the foibles of the poor biped who thinks she owns him. Such a fool, but a nice enough biped. Keeps plenty of food, knows how to scratch an ear - not useless after all.