This is a chronic problem at MAN for all airlines. For some reason MAN has the laziest and most incompetent airline staff in the UK. You should have stood your ground and insisted that they process the check-in properly and issue BP for all sectors -- which is easily done. If necessary call for a supervisor. 99% of the time I've had MAN check-in pull this stunt on me, and every time it was easy to get the BP issued (this has been the case for AA check-in as well, but that is staffed by BA at MAN). The subsequent mis-connect at JFK was inevitable, given LAN's policies.
Personally I would have just claimed from my annual travel insurance policy, which pays $500 for a delay of 4 hours or more, and made the best of the situation.