Hello, I am 'The French' I'm not a teacher,
but before you send a letter to your bank about the rate of the home loan, be sure you can change your contract or agreement you have sign.
I don't know the banking rules in India, but I do not think you can change of bank in the middle loan.
I am a saler man, and I think if you have the possibility to change, it is better to go directly to your bank to negociate face to face the decrease of your interest. If you write to the bank, they try surely to gain time.
Hope you find a way to touch your target.
Have a nice day.