Why should I have to pay credit card chargeback

I am a UK freelancer. I recently did a service job for an Australian client through a freelancer website. Before I had been hired by the client we had a skype meeting. He asked me for a ball park figure for the job. I gave him an estimate which was only an educated guess because I did not have the full details of the job. He offered to pay me that amount but I declined his offer because I did not want to work for a fixed price without having all the details of the job. He set up an hourly paid job on the website. The job took longer than originally estimated but he was charged for the hours I worked. He complained that he had been overcharged by the amount I originally gave him. He demanded that I refund him all monies in excess of my estimate. I offered to refund him part of that which he refused. He has now issued a chargeback on his credit card against the freelancer website and the website say I should pay him the chargeback amount.

My argument is that the original figure I gave the client was an estimate. It was not a quotation. A quotation is a fixed price offer that can't be changed once accepted by the customer. An estimate is an educated guess at what a job may cost - but it isn't binding. There was no contract or written quotation that said I agreed to do the job for £X. The freelancer website refuses to see my point of view and has threatened to suspend my account if I don't pay the chargeback amount.

Despite the rights and wrongs of this it may be best for me to just pay up and grin and bear it as I get a lot of work through the website and I can't afford to have my account suspended. But I just wondered what the legal issues of this are. Do I have to pay a chargeback on a service I provided just because the client says I have to. Seems unfair.