Is a suitor entitled to prevent his fiancée from leaving to attend to her necessary and private needs, even with her maharim, on the pretext that these needs will be met for her, citing the noble verse, "And abide in your houses"?

If your fiancé has not solemnized the marriage contract with you, then he is a stranger (non-mahram) and has no right to prevent you from going out. If the contract has been solemnized but he has not yet consummated the marriage with you, he does not have the right to confine you as long as you are in your parents' house, and you are not obligated to seek his permission when going out. However, if he has consummated the marriage with you, then it is his right that you do not leave his house except with his permission or for necessity. The verse: "And stay in your houses" (Quran 33:33) does not mean preventing a woman from going out for permissible matters.