BBA in Pune

MITSOM, Pune is always better than Bharati.

Not exactly for the above course but Indsearch College at Law college road offers BMS (Bachelors in management studies) which seems to be closer to what you want to study. Other colleges offering BBA are MIT, Symbiosis, etc.

Arguably, BMCC is in the top 10 of the state and should be the best choice if you stay in Pune. An average student should dream to get into this college and toppers can get in without second thoughts.
All the best.

https://www.shiksha.com/bba/course/bba-in-international-business-mit-som-college-kothrud-pune-252591
Please click the above link to get all details about the college.

https://www.shiksha.com/bba/colleges/bba-colleges-pune
Please click on the above link to get a list of good BBA colleges in Pune. Choose any college by comparing it on:
https://www.shiksha.com/bba/colleges/bba-colleges-pune5

ITM University is the best option for BBA in Mumbai.

If you want to pursue MBA then do it from any college which has good faculty. The course fee is almost 82k per year in MIT whereas you will get other ordinary colleges in and around 15k in Mumbai. Yes, it is cheaper as compared to Symbiosis. Symbiosis charges around 250000 per year. BBA is absolutely not necessary for MBA.