Given his age, I'd doubt I'd do anything. If I decided on a test that would involve anesthetic, I'd just remove it.
But just FYI: in other breeds, that location is a prime spot for lipomas (benign fatty tumours). Minnie has one on each side. We did a fine needle aspiration on one, and the vet rubbed a bit of the stuff between her fingers. It was clearly fat--even a non-vet (me) could see that. Perhaps neither of your vets thought it felt like a lipoma or saw fat.