Colts_01's suggestion is an excellent one. The international and domestic terminals at both BOM and DEL are quite far from each other (requiring a bus or taxi ride). A transfer can be a major drag to navigate even if you are not completely exhausted by several long-haul flights and a half-the-globe time change. It also sounds like you are a first-time visitor to India--given that, plus the fact that you'll be bringing the whole family, I'd try hard to keep this trip as simple as possible on your way in. BLR 's airport will be much easier to deal with.

If you change your mind and decide to spend a day/night in your gateway city before moving on to COK (which I would also recommend, just to give yourselves a chance to rest), then I'd suggest BOM as an entry point. There are a number of fine hotel choices within minutes of BOM (with DEL, you're basically stuck with the Radisson). The long flights into India and the sensory overload of arrival alone are pretty exhausting. Sitting around in the airport for another six hours waiting for your domestic flight could put all of you over the edge.