Rescued a severely infected dying cat, considering putting her to sleep

Found her on a pile of mud at our front gate. Drenched, both eyes closed, mouth inflamed, malnourished, smells like rotten corpse and not moving at all. Just trying to meow, but wihout a voice. Never saw this cat and I think someone threw her away, left her to die.

I picked her up, did a quick cleaning, covered her in a dry towel then tried to give her wet cat food. She looked starving but she didn’t even try to eat. Just kept on meowing with no sound. Still motionless.

Rushed her to vet pronto. Vet said she’s in critical condition. Female, around 6mos old weighing only 1kg. Has a big wound in the mouth. Needed to be transferred to a better facility, confined. Will require a long list of tests to diagnose

I considered putting her to sleep at that moment due to the expensive costs on an already slim chance of survival. But I couldn’t. I imagined she has only seen rough life and wanted to give her a chance of a different one. Paid it all up and told the vet to get all the care she needed.

But I’m still thinking of the best way to approach this. I still consider putting her to sleep. To give her peace. I can’t keep her anyway. Thoughts?