Is it permissible to remember Allah in the bathroom while bathing if the toilet is covered, or should one only remember Him silently due to doubt?

Worrying thoughts (Waswas) are a dangerous affliction that must be combated. Mentioning Allah by tongue inside the toilet is disliked, out of reverence for His name, except for saying "Bismillah" (in the name of Allah) when performing ablution (wudu) or ritual bath (ghusl) if it's not possible to do so outside, because it is obligatory according to some scholars and a confirmed Sunnah according to the majority. Dhikr (remembrance of Allah) by heart is permissible at all times and in all places, including the bathroom. Saying "Bismillah" for ablution is recommended (mustahabb) and not obligatory according to the majority, and whoever omits it, their purification is valid.