India

I think the first thing you should ask yourself is:
"Are you still interested in engineering after 4 years of rigorous course? "
If your answer is yes, please start preparing for GATE. Another question for yourself, "Do you want to shift career paths? "
If your answer is yes, go for an MBA. For me, this worked well.
My suggestion:
India is a progressed nation. Our education system in terms of IITs and IIMs is well known to the world. I think, pursuing MBA/M.Tech from India's top institutions is equivalent and acceptable globally. Of course, exceptions exist. If you have that financial power in access, you can opt for foreign countries as well. Education abroad is a trend these days, but it's not as glamorous as it looks. Ask any of your friend or family member who lives abroad, what they go through, and then decide accordingly.

The top hotel management colleges in South India are:
1. Institute of Hotel Management, Catering Technology and Applied Nutrition (IHM), Bangalore
2. Institute of Hotel Management, Catering Technology and Applied Nutrition (IHM), Hyderabad
3. Institute of Hotel Management, Catering Technology and Applied Nutrition (IHM), Chennai
4. International Institute of Hotel Management, Hyderabad
5. International Institute of Hotel Management, Bangalore
6. Christ University, Bangalore
7. Army Institute of Hotel Management & Catering Technology
The S.no. 1,2,3 offers B.Sc-HHA, The eligibility criteria is 12th with English and entrance test NCHM-JEE-2019.
S.no. 4 and 5 offers the following courses as:
a) Advanced Diploma in Hospitality Administration (3 years full-time course)
b) Advanced Diploma in Tourism and Hospitality Administration (3 years full-time course)
c) B.Sc in Hospitality and Tourism Studies from YCMOU (Pune Campus Only)
The eligibility criteria for above courses are 12th with English and (Electronic common hospitality admission test) E-CHAT followed by Group Discussion & Personal Interview of the short listed candidates to be held at all major cities of the country. S. no. 6 and 7 offer BHM course and conducts their own entrance test for admission.
Best of luck.