Should I speak with the suitor only out of necessity, or should I not speak with him at all until he formally proposes, knowing that he requested not to speak except out of necessity?

What the suitor said is correct, for he is a stranger to his fiancée until the marriage contract is concluded. Therefore, it is not permissible for him to be alone with her, or touch her, or go out with her except in the presence of a mahram. Your relationship is not permissible until after the contract.

The Messenger of Allah, peace and blessings be upon him, said: "No man should be alone with a woman unless there is a mahram with her." And he, peace and blessings be upon him, said: "It is better for one of you to be stabbed in the head with an iron needle than to touch a hand of a woman who is not lawful to him."

If you are harmed by the long wait, you can ask him to expedite the marriage without being alone with him or softening your speech. It is permissible for you to reject the engagement if you cannot bear the patience, but patience is better due to what you mentioned about his religiosity.