Could the doctor call your FIL and talk to him, that way you are off the hook for having to be in the middle? I think that would be my request to the doc.
or
Another way of looking at this is to put the dog, and not your family members, first. And then go forward that way. If you feel it's in the dog's best interest to be put down because it's suffering, I think that's a pretty valid point to make. Sometimes, people can have plenty of information, and they also need 'permission' to make what they feel is a horrible choice. Perhaps your FIL is coping with a lot of changes and a bit paralyzed in making a big decision like this. Likely, your FIL perceives the dog to be his best buddy and may need a nudge or more options: "You can try this medicine for temporary pain relief... or...maybe it's time to figure out how you want to say goodbye and let the dog have some peace." But I do think the welfare and comfort of this living creature needs to come before the strange family dynamics.
(Just FYI-- I have to practice what I preach professionally as well. Sometimes, there are times as a teacher that I have to tell my friends what they might not like to hear about their children-- I really try to keep the child at the center of that conversation, and any awkwardness usually blows over soon. But it certainly keeps the conversation on point, honest and productive.)