The agent was entirely correct. I presume that OP booked JFK-BOM with a connection at LHR rather than a stopover.

In that instance, without supervisory approval, the agent would have had a good deal of explaining to do as short-checking bags is not permitted. The reason for the short-check does not matter as BA's position is that you are a BOM passenger who happens to be changing aircraft at LHR.

Needless to say, the underlying sensitivity is the uptick in hidden city ticketing fraud (known politely here as xEU). A supervisor looking at this ticket would presume that OP would not have booked this ticket to somehow defraud BA.