These are both intl flights. On Air India website, look at airport information link. Specifically chose Mumbai airport.
Check-in times are different for true domestic flights and that for domestic stub of international flight.

Since 031 from BOM to AMD actually continues onwards to LHR (although as different flight numer), it is designated as international flight.
So technically, you are doing an INTL-to-INTL transfer.
But given that this is India and only thing consistent in India is that everything is inconsistent, it is best to short-check your luggage to BOM (double check that EWR agent does that) On landing in BOM, ignore any calls for transit pax and simply follow the crowd to immigration/customs. You shud not have any issues.
Now the fun begins.
1) Most likely transfer desks at arrival level (make left after customs) are open. Hand over bags to them and make sure to collect BOM_AMD BP. Then you are free to roam BOM and arrive at airport 3 hours before your flight time WITH printed BP.
2) If for some reason those arrival level transfer desks are closed, you will have to take lift to departure level (under no circumstances leave the airport building yet), hopefully, desks will take your bags and give you BP (rest same as above)
3) In rare case if neither of the above work then you are stuck with your bags during layover and NIranta airport hotel is your best bet
4) You can of course leave airport with bags and return later but that is unnecessarily painful.

95% probability that you will be fine with case 1 above. (Friends have done this successfully many times to finish off courtsey visit to in-laws in BOM and avoid having to make roundtrip to AMD to BOM during short 2-3 week visit to India)

Your dates (mid-June) can coincide with season's first heavy monsoon storm, so case 4 should be avoided. (and leave plenty of buffer if you do leave airport even with just hand luggage)