Pune

XIME is the best option for you.

Syllabus of graduation programmes is decided by the council of IITs, NITs, IIITs or other universities according to credit structure of UGC / AICTE or under which they are approved or affiliated. So, every college covers the basic syllabus of respective courses.
More or less, the syllabus is almost similar but the references used or study materials supplied are different for every college. Also, the lab activities are different.

Symbiosis Institute of Technology is part of Symbiosis International University. It is not under Pune university. You can change your branch in the 2nd year subject to rules and regulations of the university. You can change the engineering college but it's a difficult process. You may check with the registrar of the college for the process of change of branch and college.

You can get admission in Indira, KIAMS and ISBM. Indira offers MBA degree. KIAMS and ISBM offer PGDM programme.

I would suggest you KIAMS.

Right now, most of the colleges have already ended up their applications.
So in Mumbai, there are no good colleges left and same is the case of Pune.
In Pune, there is one decent college named IMDR-Pune. I think, its application is still open so check it out about this.
Best of luck.