Should I pray for the Messenger of Allah, may Allah bless him and grant him peace, between the rak'ahs of Tarawih when it is said, "Pray for the Messenger of Allah," knowing that it is an innovation, so that I am not stingy?

Praying upon the Prophet is one of the greatest acts of nearness to Allah. However, persistently doing so between the Tarawih rak'ahs is an innovation (bid'ah) because it is not ورد in the Sharia. And whoever innovates something into the religion that is not part of it, it is rejected. There is no harm in praying upon the Prophet when his name is mentioned, lest one be considered stingy. And it does not prevent the one who commanded prayers upon him from performing an unlawful act, for Malik prayed after the Asr prayer, even though he forbade praying at that time, fearing that he might be among those who, when told to bow, do not bow.