Is it permissible to complete the remaining circuits of Tawaf on the second floor of the Haram after starting the Tawaf on the ground floor?

Your circumambulation is valid if you completed two full circuits, then ascended and completed the circumambulation, because continuity in circumambulation is a Sunnah according to many scholars, and this is the preponderant opinion. According to the view that holds it obligatory, a slight interruption or an interruption due to an excuse is forgiven. Your ascent to the second floor due to overcrowding is considered a slight interruption or one due to an excuse, so it does not affect the validity of the circumambulation.