I have gone through the same situation about 5 years ago. It took me quite a few application rounds (three years) to get an offer from BCG.
When I first applied during my third (last) year at uni, I wasn't even invited for an interview. Thus, I decided to do an internship at ATK over summer.
During my masters, I applied for MBB again, and because of my relevant internship experience, I was invited to interview with McK and BCG. I reached the final round, yet could not get a job offer. Thus, I decided to take the full-time offer from ATK instead.
During 5 months, I was able to build my case around how much I enjoy consulting, and that I want to work for high profile clients and projects. And this time, I applied and was able to get an offer from BCG.
I would highly advise you to take the offer from ATK, where you will learn the foundations of consulting. And if you desire to try again for MBB, you could try after 1 year or so.
Best of luck!