I've been to Japan several times, but this time I will be 6 months pregnant. It is my younger brother's first trip and he is very excited to eat at izakaya. I don't mind a little smoke, but I don't think I've ever been to izakaya with just a little smoke.
Also, in US/Europe, it is acceptable for pregnant woman to order an alcoholic drink. Is Japanese people also okay with this? I do not want to create any discomfort.
It is still hard to find non-smoking Izakaya in Tokyo.I'm not sure this is the type of Izakaya you are looking for, but I went there few month ago and atmosphere was very quiet and nice:
Chachahana is near Golden Gai and specializes in Kyoto style homemade dishes. You can enjoy plenty of Japanese food and Japanese sake. They separate smoking and non-smoking seats. When I went there, I did not feel any smoke.
Another one is cocona:
They specialize in fresh vegetables but they also have wide range of meat and seafood menu.
In terms of culture, it is considered to be not good for a pregnant woman to drink alcoholic drinks. It is safe to go with non-alcoholic drinks if you want to avoid creating discomfort to other customers or Izakaya staffs.
I recommed Kakurebo (https://tabelog.com/tokyo/A1304/A130401/13017400/) which is along side of Yasukuni Street and on the 7th floor of comic annex of Kinokuniya book store. They separate the tables for smorking and non-smorking. The atmosphere is quiet and you can enjoy the night view of Shinjuku, if you chose the seat besides the window.