I'm guessing the 'big green pills' might have been Cephalexin 500 mg because ours are that color, but I could be wrong on that. If it is that is a common antibiotic to use.

The Clavamox is a bit more powerful of an antibiotic and is used to treat a bit different type of infection.

I'm guessing the vet that said if she wasn't spayed it would be easier would have been thinking Pyrometra (infection of the uterus) but since she's spayed that couldn't be what it is. I'm guessing that's why the first dose of antibiotics didn't help because they are somewhat guessing what type of infection it is without having a culture done which takes days to get results from.

As for the mole, it is more risky in older dogs to do surgery. Usually with older or sickly animals you have to weigh the risk of surgery vs the risk of doing nothing. Some moles are harmless some could be various forms of cancer from non aggressive to aggressive. Usually fast growth is an indicator of a cancer and would be more likely to be removed. No growth could mean it's just a mole, non cancerous growth or even a non aggressive localized cancer. Basically both vets meant the same thing they just said it differently. I think they both meant that as long as the mole stays the same size not to worry about it, but if it grows they may need to reevaluate the situation.

Just my two cents worth of unsolicited opinon. lol