Thanks for the A2A.
I honestly don’t think there are any MBA colleges worth going to in INdia which don’t have an entrance test requirement. In fact, i don’t think there are any MBA colleges in general which don’t have a entrance test requirement for new students ( students not from their own UG course ).
Even very poorly ranked colleges have a exam taking requirement. Their percentile cut offs are 60 ish but you must take the test.
Back when MICA ( a very good niche college ) cared even less about CAT/XAT/GMAT scores than it does it today, it still had taking one of those tests are a requirement.
If you’re worried because you’ve missed most of the exams for admission to 2018–2020 programs, maybe you can try your luck at MAT or MH CET.