The OP does not need a visa unless he needs to exit the airport. Staff can arrange for check-in airside. This will be much easier if not travelling with checked luggage, though it may be do-able even with checked luggage.

KE should not deny boarding since the OP is transiting and a visa is not necessary to transit in BOM. The best thing to do would be to ask them to interline the bags through to DAR at check-in in ICN.