What I would suggest you is to get More marks in physics and Chemistry so that the elgiblieliity criteria is fulfilled. 

Just don't think this as a curse, think that you failed in puc as a boon because, now you have chance to get more than 50% Aggregate . 

What if you just got pass marks , you would have not been eligible for Neet.

Now about the tricky one. Hope that the counciling process happens late and   your supplementary results would be out soon.

As of 2019. 

This year supplementary exams are early and will be completed within June 18th unlike last year where it was completed on July 8th.

The results would take aprox 20 days meaning it will be out on July 8th-14th max . 

So hopefully you might be eligible for counciling Round 2 , for national seats not sure about state counsilling  dates

If your supplementary exam results come before the Counselling results are out, then you will be eligible but if you could not show your pass certificate during documents checking, you shall be marked in eligible.