Make My Trip - Claimed amount not received

I am the F* visa student travelling to the USA for the first. I booked a flight from Make my trip. My itinerary is Bangalore (BLR) to Abu Dhabi (AUH) Alitalia by Etihad Airways.
Abudhabi (AUH) to Rome (FCO ) Alitalia by Etihad Airways and from Rome (FCO ) to Detroit (DTW) Alitalia by Delta Airways.
I got denied for carrying the * PC **+** kg check-in baggage at the Bangalore International Airport telling that they can't tag my pc check-in baggage citing that they can't tag that baggage to my last destination (Detroit) as it was operated by Delta Airways. I contacted make my trip many times long before to confirm if this itinerary will be hassle-free. They said that it is through check-in baggage. So I can collect the baggage in a final destination that is Detroit (DTW ) where my immigration will happen.
But at Bangalore (BLR ) airport they denied me and told the only way I can aboard the flight is with cabin baggage only which made me took the flight as the orientation day for my university is near.
Almost everything I have jackets clothing shoes sunglasses utility tools are back home with my parents who took the baggage home and I boarded the flight with cabin baggage of *kg. The courier has been sent via FedEx costing around ***** and other loss incurred of not having luggage at Detroit and the totalling *****. Please help in this issue to make my trip took so long in following up this issue.