Should we put our 19 year old cat to sleep?

I am very sorry to hear about your cat's cancer, as long as he is eating and drinking, and still is able to use the litter box, and plays. I would try calling a different vet they might be able to help, give it a try. I had 3 cat's about 6 yrs ago, the 1 female I had to put down because she had diabetes, she would not use the litter box she would not even eat, so my husband said that lets take her to vet and have her put down. But what I am telling you is, please try and call a different vet, he might still have a chance we try and do all we can for our pets, but the good thing is he is still eating and drinking and using the litter box and that is a good sign. I wish you all of the best for you and your cat