As you have asked about the All India Council for Technical Education (AICTE) approved BBA colleges in Kolkata, I am mentioning some of them, check below for the details

The Bhowanipur Education Society College
Address: 5, Lala Lajpat Roy Road,Lala Lajpat Rai Sarani Kolkata – 700020
Phone: (033) 22803284
Courses : - BBA

Ashutosh College
Address: 92, S.P. Mukherjee Road,Kalighat Kolkata - 700026
Phone: (33) 24554504, 24554505
Courses: BBA/ BSc Computer Science

St. Xavier`s College - Kolkata, West Bengal
Jyotirmoy School of Management and Sciences - Kolkata
NSHM Knowledge Campus - Kolkata, West Bengal
Pailan College of Management and Technology (PCMT) - Kolkata

St. Xavier’s College
Address: 30, Park Street,Park Street H.O. Kolkata - 700016
Phone: (33) 22477725
Courses: BBA/ BSc Computer Science

B.P. Poddar Institute of Management and Technology
Address: 137, V.I.P. Road,Kolkata Airport Kolkata - 700052
Landmark: Near Airport
Phone: (33) 25739607, 25739608, 25739609
Courses: BBA/ BCA

NIPS School of Hotel Management
Address: EC-98, Salt Lake City,Bidhan Nagar Kolkata - 700064
Landmark: Near G.D. Island
Phone: (33) 23586476, 23343079, 23586475
Courses: BBA

Indian Institute of Bio-social Research & Development
Address: 3A, Hindustan Road,Sarat Bose Road Kolkata - 700029
Landmark: Opposite Murlidhar Girls School
Phone: (33) 24647370
Courses: BBA / BCA

Rastraguru Surendra Nath College
Address: 85, Middle Road,Barrackpore Kolkata - 700120
Landmark: Near Cantonment Board Office
Phone: (33) 25920603
Courses: BBA/ BSc Computer Science

George College of Management & Science
Address: Budge Budge Trunk Road,Maheshtala Kolkata - 700141
Landmark: Near Mollar Gate Stop
Phone: (33) 24018687
Courses: BBM

Netaji Subhash Engineering College
Address: Garia Panchpota,Kolkata - 700152
Landmark: Near Garia Station Road
Phone: (33) 24368888, 24364926, 24364929
Courses: BBA / BCA