Is it permissible for a Muslim to recite the supplication for looking in the mirror or other forms of remembrance (dhikr) inside the lavatory?

It is disliked to mention Allah Almighty inside the lavatory, whether it be Quran or supplications, out of reverence for Allah's sanctity. This includes all forms of remembrance and speech, except for speech necessitated by need. This dislike is one of tanzih (disliking that which is better to avoid), not tahrim (prohibition).

If one sneezes, they should not praise Allah with their tongue but with their heart. The same applies during intercourse.

Similarly, some scholars deem performing ablution inside the lavatory disliked. However, if one performs ablution inside it, the preponderant opinion is that they should not recite the remembrances of ablution, out of reverence for the remembrance of Allah Almighty.