I have an uncle who gets extremely annoyed when my dog sleeps under the Shabbat table. Is it a major prohibition that the dog not be in the room during the meal/blessings? May I never say a blessing in front of the dog even on a weekday?

It's not a major prohibition to have your dog under your dinner table, but it is a major prohibition to make your uncle, or anyone for that matter, extremely annoyed. Try to reach a compromise on that issue – it's not worth fighting for. Kol Tuv