There seems to be 2 open spots at QB for Rattler and Haener. DA’s statements about them after the draft indicates both can or will have a roster spot on the 53.
My OP suggested that the Saints could go either way with this based on how they perform in camp and preseason. But yeah, it's possible the Saints carry 4 QBs on the 53-man roster. I covered that possibility in the thread titled " Is The Saints QB Room Set For 2024". Both Haener and Rattler would have to shine in camp and preseason so much so that the Saints wouldn't want to take a chance on cutting one of them hoping that they would clear waivers and could return on the PS.