@Flying_Knight
Looking at the current regulations and related CAR, is is highly recommended that, you do your recency in India, you shall be prepared to fly for any were around 25hr. since you have SE commercial pilot license, i don't think you need to appear for any Tech specific exams. however i would recommend you to see DGCA, for the fact that the flight training was done in mixed aircraft fleet(Arrow,C-172, Cherokee) so that if any shortfall is observed by the DGCA officer you could get them adjusted in your recency.
You would need a SPL and FRTOL from DGCA which could only be helped out by a flight school.
I am not sure with karnal flight school. but would highly suggest you to choose a flight school that is operated by a CFI, in india some flight schools are also operated with PI or PII, though PI and PII can finish your recency, you couldn't be able to do the checks, that are only supposed to be conducted from a CFI, and Yes these "checks" what i mentioned before shall satisfy DGCA and your first point of "flight reviews test" and shall not have any problem.
5 hr of multi engine didn't make sense to me, either you give 3 checks(Day, night, IR) in a ME that is endorsed on your FAA license(Endorsed as per DGCA could be understood as at least 10 flight hr done on that A/C and checked or preferably given check ride on it) or you fly for full 10hr. Since you said you have a FAA ASEL CPL and not AMEL, i am not sure with your specific requirement.
I am not quite sure but i don't think DGCA checks with your flight school about those 300 hours, but yes for sure they check for where you did your recency, So if you do your recency from India you shall be good. but how ever they will sure ask you the flight school approval letter(Air Agency Certificate) during you period of training from the flight schools, which is often available on the flight school website and also preferably training has to be done from a PART 141 flight school.
This is the best i could help you.
cheers.