Lol. No, not at all. That is not how ATC or FIRs work. No one needs any permission from anyone to enter any FIR. All Mumbai ATC does is provide flight information service and an alerting service to commercial and private aircraft as per the international agreement through the International Civil Aviation Organization. The airline pays the Mumbai ATC for its service. That is all an FIR is.
Otherwise, it would be mighty embarrassing for you guys since we regularly fly our missiles and military aircraft in Mumbai's FIR.