And the reason they don't show availability when adding and 3rd person is because the rooms are incredibly small and cant accommodate a 3rd person so you should call the hotel to inquire about the infant.
I get that I have stayed there multiple times, I just feel like (maybe unjustifiably), if the infant is not taking up any significant amount of space (sleeps with parents, or in small space) and does not eat, why then is it the hotels concern if they are present?