Joanna, we are in the business for over 25 years and work closely with international clients.
The Sony HVR-V1E camera is small enough, BUT wouldn’t pass as being used for amateur hobby filming.
The need for permits will be reliant on the areas where you wish to shoot. It is not essential that you need to furnish a script for everything you film in India.
My advice would to be enter into a dialogue with the Indian High Commission closest to you and tell them you wish to shoot an international travelogue in India. A travelogue does not require a detailed script, so that allows you to shoot just about anywhere except areas of sensitive nature from the commonsense as well as the Indian government point of view. When you make such an application to the Indian High Commission, they will be forced to write to you. This way you are on record as having approached the authorities. Let them tell you whether a permit is required or not. (IVI/1)