You take a cat which is capable of running away as you said, into the most beautiful wilderness you can find and you leave it there.. This means the cat experiences living alone properly in absolute choice of where to finally lay and die, not only that, goes to the calmest, rightest spiritual place for an animal.
Or you may have always petted it, then you plug in the carbon monoxide exhaust from a generator in a box on your knee, say your goodbyes and let an OD of sedatives take the pain of slowly leaving.
Then you bury her/him shallow, quite shallow and put a few rocks on to keep the animals off and allow that same return to wilderness.
Or finally you let nature take its course and maybe you will be there right at the end, maybe not. Sedatives are good.
In a sane world however, death would be naturally sedated. Final resting place would preferably be the ocean, it is volatile with energy and spirit, particularly the pacific.
You could invite the chakras into your teepee and develop a cat like accord with your spirit animal though... that might be quite the hunt, even according to medical science.