I don't think any of the options stated above are worth it. They will consume more time, and cost you a lot of money and effort. So rather than partial drop, I recommend you only drop the year and fully concentrate on the JEE syllabus. Now you have JEE exams and NEET twice a year which gives you more chances of cracking it. I have listed 4 probabilities and their possible consequences:
Not dropping a year and studying in an NIT and preparing for JEE along with it and not being able to crack it is Waste of time and an excessive study load.
Studying in an NIT and preparing for JEE along and cracking it: Only a waste of money
Fully dropping a year and concentrating entirely only on JEE and not being able to crack it is Waste of time but you are lucky enough to give a second shot the same year.
Dropping a year, fully concentrating on cracking the JEE exams, and cracking it: Your time and efforts are well invested.
The ultimate choice is yours. But dropping a year is better than a partial drop I guess.