Is it the carrier's responsibility to bringing a passenger home?

Short story

My girlfriend is currently on a business trip in India. Due to issues with her reservation, she is concerned she might have troubles checking in for the plane back home. However, seeing as she is mid-itinerary and on the return leg of the journey, is it the carrier's responsibility to bring her home, similar as to a passenger being denied entry to the destination country?

Longer story

Her trip is booked through a travel agency (company policy). The carrier is Lufthansa, but the operator for the first leg of the return journey is Air India (and later SAS).

Before departure, she noticed her ticket doesn't actually carry her exact name as in the passport but an abbreviated form (example: Kate instead of Catherine, except her name is uncommon in the English-speaking world). She was advised that the ticket could not be changed, so she would have to take her chances.

She did get to India, however presently she is not able to check in online for the return trip, as her reservation does not exist in the system (using the booking reference or any version of her name).

Lufthansa says they cannot help further, as the reservation was made through a travel agency, but points to the operator for check-in. The operator (Air India) doesn't have her booking.

Normally, this would be an issue that should be contested at the airport, however apparently there is currently a security check at the entrance of the airport, and you will not be admitted without a printed ticket matching the passport.

The travel agency has advised her that she should either buy a new ticket, or take her chances that they will magically find her reservation at the airport.

They also say that in the event of problems (reservation cannot be found, or she is denied boarding due to naming issues), she should argue to the carrier that it is their responsibility to bring her back, as it is a return journey.