BOM airport inside, is pretty clean and save, but otherwise it's a pretty grim place to spend any time in. As others have mentioned, even the business lounges are incredibly basic; uncomfortable seats with very limited food and drink. Also they are airside so you may not get access until the AF check-in opens and you can get an invitation.

The most common recommendation I've seen is to go to one of the 5* hotels in the area, and arrange for them to pick you up and drop you back to the airport as well. They are very expensive, but they are also very nice.

This also has a big added benefit of making your transfer from the domestic to the International terminal much more comfortable as well, which otherwise can be a bit stressful.

I think Savage25's suggestion sounds like a very good one, otherwise I believe some of the hotels will do a half-day rate for passengers departing on late nite flights if you contact them direct. This could still cost you 7000Rs + with transfers though. There are also booking agents in the terminals at BOM that you could negotiate with (when you arrive,) for a better deal at one of the 5star hotels. Worth trying if you don't book anything.

I'd personally would hesitate about heading down into Mumbai unless you are very confident about dealing with the taxis and stuff. There are pre-paid taxis which can drop you into Mumbai but you might end up having to haggle on the way back. It is also quite a long way and can take 90mins + each way if the traffic is bad.