Mumbai

Please refer to the link provided below for an exhaustive list of colleges which offer courses in B.Sc in Mumbai. You can use the filters on the web page to navigate to a college of your choice. Additionally, you can also download their brochure which contains details regarding admissions, course eligibility, etc.
https://www.shiksha.com/science/colleges/b-sc-colleges-mumbai-all

Sp Jain University is one of the best college who have international programme.

Please refer to the link provided below for details regarding admissions in St. Xavier's College, Mumbai for B.Sc in psychology. Download their brochure which contains all information regarding course eligibility, admission criteria, reviews and rankings, fee structures etc.
https://www.shiksha.com/science/course/bachelor-of-science-b-sc-st-xavier-s-college-mahapalika-marg-mumbai-110115

You should consult with admission authority of your college whether you will be permitted to transfer to another college midway through your course. Mostly, lateral transfers are permitted for students who have a good academic record. Also, check whether your college has a tie-up with the other institute. Make sure you block a seat in the target institute and then give up your seat in the current college. Good Luck!!

I did not do BMM but I stayed in Wilson College girl's hostel and we had a good number of people who were doing that course. They loved it in terms of the faculty, the industrial visits (IVs), the exposure that a media student should be getting, and definitely the classroom culture and crowd. Based on their experience, I would definitely suggest you to go ahead with this choice.