According to this book[1] war rockets were used in India at least as early as the 14th century, and mughal emperor Akbar used metal-headed ones in the 16th. So it would seem neither rockets nor metal ones were invented by Tipu. However it doesn't answer your question of who did invent the war rocket :-)
But IIUC what you want to who under Tipu's rule invented (or revived) this rocket. You're probably right in thinking it wasn't Tipu himself who did this, since he was a king and not a scientist. But i suspect the real inventor's name is lost to history.
1. The Mughal Empire at War: Babur, Akbar and the Indian Military Revolution, 1500-1605 (Asian States and Empires) https://www.amazon.in/dp/B01EX69NJ0/ref=cm_sw_r_apan_glt_DMX...
Edit: added the link