You don't have to clear immigration at all. After alighting at Bombay you will be bussed to the arrival gate where as you enter look on the right for domestic arrivals. Keep your boarding pass handy. And keep in front of immigration desk. One stamp on the boarding pass without he as much as looking at your face. All he needs to see is the big D on the boarding pass.

Walk to baggage claim, collect bags, don't stand in queue for baggage screening and walk through the green channel showing your domestic boarding pass to anyone who cares to look.

The only id you need on the flight is id which is acceptable to enter airport in Goa and at check in if asked. Similar to domestic flights.