Changed Lufthansa flights several times both online or with the help of very friendly and competent call center agents.
Never a problem to get it free of charge but I am flexible and could search for a flight which wasn't more expensive than the one which I had booked initially.
Don't know anything about your particular flight, but it's very important to understand the difference between a change fee and the actual fee for your flight.
All airlines which offer a flight change free of charge will not charge you an extra fee for doing so BUT all of them will make you pay for the flight fare difference.
You might have booked a very cheap discounted flight and the flight you have chosen now is for some reason very expensive.
Always worth to check the current flight fare before asking for a change. You have to find a flight which is not more expensive than the flight which you originally booked , otherwise you have to pay the difference.