Please refer (https://www.nirfindia.org/2020/Ranking2020.html) for latest rankings.
Whenever people have questions as you have posed here college versus branch, there is not a single objective answer to this. Response varies from people to people some prefer college over branch on other hand some give preference to branch.
Your Alma Mater matters a lot in your future course of actions in your career, IITs have strong Alumni Network, grads passed out from these premier institutes have made significant contributions to Technology, Engineering, Leadership and being part of such a Network helps an individual in every phase of his life.
CS from atleast top-7 IITs is an achievement and for that one need to be a top ranker in JEE competitive exam. Getting a degree from these institutes provided you maintain your GPA score can place you to best Technology companies with competitive compensation.
BITS is also among better institutes in the country, much better than 99% of engineering colleges in India in terms of pedagogy, placements, infra, alumni network etc. They are like second to none.
However given a chance IIT is an experience, a way of life, if you make a seat there it will remain for your lifetime and not just for 4 years of your path to getting engineering degree.