We haven't been there yet (hence this thread) but my research suggests that August-September will not be good. For example, 2017 saw Mumbai flooded:
2017 Mumbai flood - Wikipedia
Also, check this site:
Best Time to Visit India - Peak Season for India - When to Visit India - India Travel Season
After reviewing lots of sites & comments, I came up with this overview of when is GOOD (but no actual experience to go by):
September: Sri Lanka east coast
October: Simla, Darjeeling, Delhi, Agra, Jaipur, Kolkata.
November: As for Oct plus add Amritsar, Sri Lanka West (but better from Dec)
December: As for November plus add Maldives & Kerala and leave out Simla.
January: Same as December but getting quite cold in Amritsar and forget Darjeeling.
I did not look at Feb through August.
Note that this list is only based on weather and does not consider hotel prices, etc.
Also, there has been political unrest in Darjeeling as noted earlier in this thread.