DoubtRama never had any doubt. Even layman's doubt was already long-ago cleared by Agni Pariksha. People thinking it was some easy drama. Just put your hand on top of candle for 5 seconds, and see if you can bear it. Only people who have done agni pariksha themselves have right to comment about Sita devi or Draupadi (who does Agni snanam before entering house of other husband).
People keep on questioning all the time. These people will also question Rama as why he killed Vali from hiding. Vali himself asked Rama 6 questions, and after Rama satisfactorily answered, Vali gave his son Angada to Rama for guardianship. Still, even today, there is debate, because they don't listen to Ramayana upanyasam from elders fully.
Rama wanted his wife who was garbhavati, to be under Rishi's care. Because during garbham, all good things must be heard by embryo/foetus. And who better than Valmiki muni to impart this knowledge. And after they are born, Lava & Kusha learn Ramayana from Valmiki.
Also, as a king, and as Dharma incarnate (ramo vigrahavaan dharmah), he is not supposed to ignore his subjects, however varied their opinions might be.
So, as '2 mango with 1 stone', he sent Sita devi to Valmiki ashrama.