Being a civil engineer my self, i need to tell you that the interior designing is much close to architectural side than to civil engineering side.
To elaborate civil engineering basically deals with strength stability and design of structures whereas architectural side mainly deals with looks beautification, space utilization and by laws related to structure. If you want to have carrier in interior designing, let me tell you now a days it have great scope and this field provide you ample chance to use your creativity and talent.
The good colleges in Mumbai as per my knowledge are
1 Sir JJ School of arts
2 amity university
I am not pretty sure about fees as it purely depends upon college to college but it can be anywhere around 40k to 80k per semester if you are going for good college.
My suggestion will be,
Look for entrepreneurship rather than a job in this field
Go for short term course instead of masters or complete degree
Study vastushtra and feng shui as they are in high demand specially in India.
If you want to go for job you can have a career in banks as they recruit planners and interior designers on contract.
You can also go for both the furnishing works and home decoration along with interior designing.
Best of luck!