Is it obligatory to complete the Tasbih after the obligatory prayer before performing the Sunnah prayers, or is it permissible to perform part of it and then pray and complete the Tasbih later, especially when time is limited?

The Prophet, peace and blessings be upon him, forbade connecting one prayer immediately with another until we speak or move from our place, and this is for recommendation ( مستحب ). The original principle for theذكر (remembrance of Allah) after prayer is that it should be immediately after the obligatory prayer. If you wish to quickly offer the supererogatory ( نافلة ) prayer, you may say some of the adhkar or move from your place, then complete the rest (of the adhkar) after the supererogatory prayer, provided that this does not become a regular habit.