Flying without a doctor's note

So, I'll be flying when I'm 27weeks or less, but I am overweight and wouldn't want the airlines to think I'm further along than I really am. I asked my doctor for a note that I could carry with me while I travel and she wouldn't give it to me. I don't know if she just thought I wouldn't need it or what. She told me I'd be fine to travel and gave me tips, but wouldn't sign a note to clear me if something comes up.

Anyone fly without a note or have advice on if I should insist on getting one?