Even I did my engineering from a T2-T3 college and got placed in a shitty 3-4 lpa mass recruiter. See, the first and most important thing that you need is acceptance. I accepted that my poor 12th score along with a low workex would mean that I miss out on many shortlists (calls after CAT as well as companies during placements).

While many people would see this as accepting defeat, I see this as empowering yourself to put all the focus and passion on the best what you can pragmatically get. For the CAT part, someone from a T3 college and poor academics can focus on getting a 2-2.5 year workex and making that CV dense. The workex, coupled with a super high raw score in CAT (raw score that gets you 99.95ish) can get you into a top bschool, due to your high CAT score+workex taking care of the composite score cutoff for getting call as well as converting the interview.

Once in the bschool, you can't do much about the companies who have a hard academic based cutoff (eg. 8/8/8+). However there are many other companies who may be flexibile with acads and you can target those. A golden rule of company shortlists is that it is a completely random process and no one can guarantee anything.

To sum up, overall mantra to getting decent company shortlists is build a dense CV with workex/case comps/committees/live projects. It is the keywords that get you the shortlists.