Is it permissible to remember Allah and recite the invocations for ablution while at the washbasin inside the bathroom, given that there is a plastic partition between the washbasin and the toilet seat that does not extend to the ceiling?

There is no harm in performing dhikr (remembrance of God) and tasmiyah (uttering "Bismillah") in the place of ablution (wudu) and ritual bathing (ghusl) if it is separated from the area of relieving oneself by a partition. Sheikh Ibn 'Uthaymeen permitted this as long as the washing facilities are outside the area of relieving oneself, even if they share the same roof. He affirmed that there is no harm if the two areas are separated by a partition or a door. However, if the toilets and the place of relieving oneself are in the same area as wudu and ghusl, Sheikh Ibn 'Uthaymeen holds that uttering tasmiyah verbally is permissible, but it is better to perform tasmiyah in one's heart without verbalizing it.