Is it allowed for a woman to wear a gold necklace with Ayatul Kursi at all times, including when using the bathroom or taking a shower?
Bismihi ta'ala,
Yes, it is permissible for her to wear it, as long as the script of the Quran does not touch her skin if she does not have wudhu'. To avoid this, she can wear it on top of clothing, or keep the script to the front, or cover it with a thin transparent layer of nail-polish, or something like that, which can also be used as a protection for it as well.
Of course, she must also ensure that it stays away from any najasah.
And Allah knows best