Well, I am not a vet and I doubt I can be of much direct help for your specific case.
However I did have a 17-year old cat w/ a very fast-developing fibrosarcoma on one of her rear legs, right above the foot. My advice from a veterinary oncology specialist was pretty much the same as you got. Apparently fibrosarcomas are very aggressively locally invasive ( and in my poor cat’s case it was astoninshing how rapidly the damn thing popped up and grew ), however don’t tend to metastasize remotely, so surgery with as wide an excision zone as possible is usually the best treatment. With my pet the only really good treatment ption was amputation - the tumor was all into her foot and tendons and trying to just excise the tumor and treat w/radiation, in addition to being more expensive and probably stressful for her ( at 17, I didn’t want to do anything that would make her miserably ill for the rest of what might well be a short life at her age ), was a lot less likely to be successful. So after some agonizing, assurances from multiple vets that cats generally adjusted quite well to a three-legged existence w/reasonably good quality of life and her blood work and x-rays ( showing her spine and hips were in good enough shape to bear the strain ) showed she was up to it, I had her leg amputated at the hip. As it turned out she did indeed recover quite nicely and very quickly. Unfortunately she died several months later from a heart attack while coming out of anaesthesia ( she was in to have an abscessed tooth removed ), so I can’t really say if the treatment was definitive, but my understanding is that such surgeries tend to basically fully curative when the tumor can be removed so thoroughly.
A tumor on the side I imagine it is a more difficult situation, as some ribs may be involved. 13 isn’t as ancient as my cat was, but is still technically geriatric, I believe. I’d be inclined to follow whatever treatment had the best combination of a) the best chance of success and b) whatever will give her the best quality of life for her remaining years. In my case, I wouldn’t have pursued chemo or radiation for that latter reason. If surgery hadn’t been an option, I would probably have had her euthanized. But every circumstance is different.
Whatever you decide, best of luck :).