Yes you can. Check out blogs in my Tripoto profile about places to visit in South India, Most of them are Karnataka and Tamil Nadu. Let me know if you have any questions
Yeah definitely ,a month is sufficient to complete those 3 states.
segregate the best Kerala:- munnar,allapuzha,kochi,wayanad,
tiruvanantapuram, athirpally waterfalls
tamilnadu :- ooty, Chennai, pondi,coorg,Thanjavur
and search more for ur basic interset
From Mumbai plan as follows,
goa, Karnataka, places r as follows,
gokarna,
karwar, murdeshwar,
udupi,
Mangalore, enter Kerala,
kochi,
alleppey,
munnar,
attirapally water falls,
varkala, trivandrum, enter Tamil Nadu,
kanyakumari,
madurai, rameshwaram,
dhanushkodi,
velankani, kumbakonam,
tanjavoor,
puducherry, mahabalipuram,
Chennai then to bangalore, nandi hills n near by places n thngo to coorg , madekeri, Mysore n hampi n hospet, dhudsagar falls n then bck to ur place. try to include Ooty n coonnur in Tamil nadu