You have to do the Integrated BA LLB program but you can't do BA and LLB simultaneously because, for LLB you need to pass the BA. As a result you have take admission to the 5 years BA LLB course.

Now the eligibility of the UPSC CSE is that the student need to pass the Bachelors Degree in any field or stream. so yeah! if you wish to go for the UPSC CSE then you are eligible, but you can't do 3 years BA and 3 years LLB simultaneously.

For more information regarding BA LLB consider the link below:

https://www.careers360.com/courses/ba-llb-dual-degree

I hope this helps. Wish you a great day!