I would say the best route would be.
Kottayam - Palakkad - Coimbatore - Avinashi - Salem - Bangalore (overnight stop) - Davangere - Belgaum - Satara - Pune - Mumbai.
Kottayam - Palakkad - Coimbatore: I am not sure about the road conditions / route. Joining NH 544 (Salem - Kochi Highway) at Angamaly is suggested by Google Maps.
Coimbatore - Bangalore : I have been on this route just last weekend. Can vouch for it. Amazing roads.
Bangalore - Pune - Mumbai: Should be good as well.
Bangalore Stay - You have lot of options. If you are planning an early morning start, you can stay inside the city. Any start later than 730am, you are better off staying along the NICE road entry/exit points. You can stay near Electronic City, for an early morning start.
If you have more than 3 days, you can do the coastal, scenic road as well. But it will be tiring with a lot of attractions to see (if with family).