Is it disliked or forbidden for a suitor to ask his fiancée about her prayer and fasting, and are there specific manners to prevent him from asking this embarrassing question, especially when there is a legitimate شرعي excuse preventing her from praying that he does not understand?

If it is merely an engagement, then the suitor is a stranger to his fiancée, and he does not have the right to speak with her except out of necessity and while observing the Sharia (definition) guidelines. There is no need for him to ask you specific questions daily, and if he has concluded the marriage contract with you, then you are his wife, and it is permissible for him to speak with you. However, he should not ask these questions as long as you feel embarrassed by them, and you can hint to him that these questions are embarrassing to you without stating the reason.