Hey Sourav,

As Mechanical Engineering is the core branch you can shift to any of the sub braches in your specialization But Production engineering is a specialization subject of Mechanical engineering. if you are from mechanical engineering stream than apply GATE from Mechanical. As if you will get a general rank below 600 air than you will have an interview letter from very reputed PSU public sector unit and also you have an option for M.Tech from IITs. While if you will apply GATE from production stream than you have only 1 option i.e. Mtech. This is the best way also to get M.Tech in IITs. But if you are a production engineering graduate and you are going to apply for GATE than after getting a good rank you have either 2 or 3 PSU and M.Tech from a good institute. 

Hope that helps all the best