If one doubts during prostration in the last Rak'ah of the 'Ishā prayer whether they performed the Rukū', does this necessitate repeating the Rukū', completing the prayer, and performing the prostration of forgetfulness (Sujud as-Sahw)? And is this action correct, and is the prayer considered valid?

If a worshiper doubts having omitted a pillar (rukn) and has not yet reached its place in the subsequent rak'ah, they must return to perform it and what follows it, because doubt about omitting a pillar is treated as if it were omitted.

Therefore, you were obliged to return to bowing immediately upon doubting whether you had performed it. Your prostration while doubting the performance of bowing was superfluous.

However, since you performed the bowing and the subsequent pillars, your prayer is valid, and this unintended addition you made out of ignorance does not harm it, provided that the doubt was not a mere illusion or a form of waswasah (obsessive whispering).

If you are prone to waswasah, you must disregard these whisperings and pay no attention to them.

Shaykh Ibn 'Uthaymeen said: "Whoever doubts having omitted a pillar, it is as if they have omitted it. If the doubt occurs during the prayer, and it is a genuine doubt, not an illusion or waswasah, then if they were prostrating and, during their prostration, doubted whether they had bowed or not, we would tell them: 'Stand up and bow,' because the default is that the bowing was not performed."