Can I pursue BBA along with Aviation as an integrated course?

Why not? But, I suggest you to keep your prime degree as full-time and the later as part-time of through recognised distance education wing.

UPES-Dehradun, Osmania University, Banasthali University, Aptech Aviation and Hospitality Academy, Sun Institute of Technology and Management-Vishakapatanam, AJ Aviation Academy, etc. offers similar courses.