Can’t speak for BOM-HEL-MXP in particular, but for DEL-HEL-various other Schengen airports (and also for DEL-HEL-US) on AY, in India I have repeatedly seen AY check in and allow travel to US passport-using and Indian passport-using travelers to fly when the ticketed name order is opposite of that which may be suggested by passport swipes. In such cases, the check-in at the airport will have to be done by check-in staff pulling up the PNR by typing in the record locator and/or the last name as on the ticket instead of as per a passport swipe.

Will CISF make an issue about letting a person into the terminal building to check-in when the name order appears switched? It may depend upon what is used for identification to enter the Indian airport.