It's so hard to know what to suggest. I have experienced both sides of the coin in the last few months. In one case, my cat Charlie who seemed in very good health and around the same age as your Mr. Meow (aged 6) had a golf ball sized lump on his side (that seemed to appear out of nowhere so I think it must have been fast growing) underwent a fine needle aspiration to make sure it wasn't an abscess or a fat deposit. It was neither. The vet wanted to put him under and take a biopsy but I asked him to take the whole thing if he was already going to be under. Charlie, died just after the surgery but the vet believes Charlie had cardiomyopathy which was unknown to everyone. The thing is though that the vet said the surgery itself went so well and that he managed to excise the tumor completely and he said he got all the surrounding tissue as well. Lab results later confirmed this. He said he was sure that Charlie would have gone on to live a very long life.
My other cat Tiger had a tumor removed from his nasal passage a few months ago as well at the age of 11. He wouldn't be here without the surgery because he was in terrible shape at the time so I'm so thankful we went through with it.
I think a fine needle aspiration is a good idea, they can do it so quickly and Charlie didn't seem to feel any discomfort when they did it right in the exam room. Perhaps there is a way to determine if Mr. Meow has any issue with Cardiomyopathy prior to anesthesia? I think if Mr. Meow is healthy otherwise and strong enough to handle the anesthesia that it might be a good option. But again, it's so hard to say what is the right thing. They're so important to us and these are very big decisions.