About 3 years ago I had a similar situation with Delta. I spent the time in Terminal 1 and was generally bored - it's not an exciting terminal. I looked closely for an exit to Japan and couldn't find one. Even with access to the ANA F lounge (which I've been too) I'd enter Japan upon arrival from BOM. The pre-security area of terminal 1 is already an improvement.

I haven't been, or researched it recently, but what about entering Japan and heading to Nine Hours, the capsule hotel in T2? You can get just a shower there. It may not be super convenient but less of a hassle than convincing someone to escort you to immigration.