You can post your query on 'Flats without Brokers in Mumbai' group & page on Facebook.. I have seen people give out full apartments for shorter lease. Though, such options are few.
Alternatively, why don't you try Airbnb? I have seen many nice apartments available by the day - week - month. Depending on which area in Mumbai you are looking at, it might work out to be similar or lower than the average rent in that area. about 5 years ago