Is it permissible to propose to a close relative who has memorized the Book of Allah, while fearing falling into the forbidden or harming her?

Yes, what you did by not speaking to her was correct, and beware of doing so, for that is from the footsteps of Satan. Therefore, we advise you to propose to her as long as she is pleasing in religious commitment; based on the Prophet's (peace be upon him) saying: "There is nothing better for two who love each other than marriage," and his saying: "O young people, whoever among you can afford to marry, let him do so, for it is more lowering of the gaze and more protective of the private parts. And whoever cannot afford it, let him fast, for it will be a shield for him." If you are unable to propose to her now, then adhere to patience and chastity.