Sounds strange. You cannot benefit from a cheaper fare, to pay less, but it should be the same, especially if it's recalculated properly and I am not aware of rebooking fares vs new fares online. You might be a victim of LH agents who make stuff up.

Could you post the fare calculation from your original ticket and the dates you are looking to change to?

Of course, if LH had vouchers you'd be able to get around this and maybe even skip needing to call-in.