So sorry to hear this. When we have pets, especially dogs or cats, they become like children and part of the family.
First off, you use traditional charts, I don't, so I hope you don't mind if I redid the chart to include the outer planets.
Your cat as you wrote is the 6th house, sagittarius ruled by Jupiter.
South node is there showing he is suffering.
On my chart I entered the Fixed star Sinistra, as you see it is in the cat's house, and almost conjunct south node. This star represents mortality by poisoning.
So this gives us the first clue. Your cat has some sort of intoxication.
The 6th from the 6th, which is his illness, is the radical 11th. Ruled by taurus we find sun and mercury here.
Taurus in general is over indulgence and also recuperative forces, which in this case are important given the recent surgery.
Sun is excess. So we have too much of something which is poisoning him.
The cat is ruled by Jupiter, ruler of the 6th, and Jupiter is located in the cat's 2nd house. The 2nd house rules nutrition. Jupiter in Capricorn is weak, so again pointing to a nutrition problem.
Taurus rules the throat and larynx.
The 7th of the 6th is the doctor. Here we have empowering north node. However gemini on the cusp indicates more than one doctor, with Venus there separating from a square with Neptune, liquids, as well as poisons. It looks as though one of the doctors made a misjudgement with medications, which is having a bad effect on the cat.
You are worried about the cat dying, which is a natural concern. The cat's house of death is the 8th from the 6th, which is the chart's ascendent (and you), ruled by moon. Moon is in 5 degrees sagittarius, in the hospital, but doesn't make any significant aspects until it squares Neptune! Neptune is in the cat's 4th house, home, ruled by pisces, water.
So what is this liquid that is having a bad effect on him?
Mercury represents his doctor (ruler of the cat's 7th) and mercury is conjunct the 6th of the cat's illness. Mercury however approaches a trine with Jupiter, the cat, from the sickness house to the nutrition house. He can find the problem and the cure. It might be something as simple as changing his medication.
Jupiter is also the liver, which is being effected by this liquid that is poisoning him.
Taurus is ruled by Venus, who governs digestive disorders, another confirmation.
Further, the cat's 2nd house of nutrition is ruled by Saturn in Capricorn, Saturn rules poisons.
The 2nd of the cat also in general rules the tongue, heart and stomach.
Moon in sagittarius could also point to the liver, or a growth, or tumor.
Talk to the vet and hear him out about what is being given to the cat.
It's too early to talk about putting him to sleep.
Hope this is helpful, God bless.