Your diagnosis of the context (India) in general and of Uttar Pradesh in particular is spot-on. The rise of radical (or militant) Hinduism in the recent past is very much palpable across the nation. However, in the midst of this change, the positive thing is that there are a few dissenting voices from within the Hindu fold that is challenging the radical view of the given majority. For example, last year (2018) Dr. Shashi Tharoor published a book titled as “Why I am a Hindu” wherein he not only gave reasons as to why he is a Hindu but he also distanced himself from the radical hinduism. We would do well to read this book and others too in order for us to understand the nuance of Hinduism.
Anyway, coming to your question of how do we engage meaningfully with our neighbours who are predominantly Hindus. Before I do so, may I also encourage you to kindly read my other responses to Kishore and Carson about some of the general things, I have included on how to build bridges with others.
Given below are some of my suggestions:
Begin with something you can agree with. For instance, unlike an atheist who does not believe in God, I would agree with a Hindu because He believes in God (although, different from the Triune God). Similarly, a Hindu believes not only that God exists, but that God is ultimate, infinite and above all God is active in this world. Of course, in our discussion we should eventually and gently try to draw the distinction between the two world-views.
I am sure, you would have also noticed by now their deep desire or hunger to relate with their respective deity (which ever they chose to embrace). Furthermore, they also believe in the law of Karma and somehow deep inside craves for liberation (or moksha) which is salvation from the Christian point of view. Hence, their emphasis is in doing the good while on the other hand some would recommend yoga, the path to ultimate liberation. However, apart from the long duration of time that it requires, there is also no assurance whatsoever that one would ever attain success.
Now, at this point, we should highlight the distinctiveness of the gospel of Christ where salvation is not by works but it is a gift of God and not by works. Above all there is assurance of salvation too, based on what Christ has accomplished for all of humanity.
Most importantly, while witnessing you must also pray that God would intervene by opening their heart to what you are sharing. All we can do is to try to convince them, but conviction comes from God.
I would also like to encourage you to attend our two weeks training program which takes place three times in a year in Chennai. Through this program you would be further equipped in the whole discipline of Christian apologetics. Just incase you are interested you can reach out to me at email@example.com
I would highly recommend that you read the book, “Jesus talks to Krishna” authored by Ravi Zacharias. It is an excellent book that would give you more insights on the subject.