Can't speak from first hand experience of owning a ":tripod" but there are many three legged bassets that do just fine. From a purely technical perspective the basset carries most of its weight on the front end. the hind end as relatively light in terms of weight bearing so it is for them to deal with a rear leg loss though there are plenty of mobile basset with a single fron leg as well.
We had a basset that developed a tumor on the inside of her back leg around the age of 8. It came on very quickly. The Vet did a biopsy, which showed the tumor to be non-malignant. We chose not to have it removed b/c it would have interfered with the movement of her back leg. She was very active, jumped on and off all the furniture in the house, walked, hiked, swam. The tumor grew to be the size of a small tangerine. It never seemed to cause her any problem. We lost her this past fall to what the Vet called a "big dog" disease, mass on liver and spleen, totally not related to the tumor on her leg. Four weeks before she crossed the rainbow bridge, at age 11, due to the liver problem, she hiked with us five miles up hill! She was a happy bassett with a leg tumor! If we had to do it over, we'd choose the same option, not to operate on the non-maligant tumor.
What kind of tumor is it? If it's not malignant and not imparing movement, I'd leave it alone. My girl Penny had a mast cell tumor (malignant) and it took two surgeries to get it all. She's went through a lot, but it was totally worth it, as there was really no other option. I really hope in your case the situation isn't as bad as having to amputate, but I've seen many "tripods" getting along just fine. Have you thought about going elsewhere for a second opinion?