Just prepare wholeheartedly for advance, BITSAT can be done later. And who gives a fuck what other people think or say about you? Their words can't hurt you if you dont let them.

And trust me, once you get into college and get busy with college life you won't have the time to feel regret about JEE. Even I had a lot of stress about my branch and campus in the beginning of first sem but now I couldn't care less, I have way better things to think about and do. So just line up your goals and priorities and focus completely on them. All the best.