I did this a week or so ago, but on the phone.

I was trying to change both route and cabin, and the phone agent suggested that it would be easier to canx the original booking and start again. After consulting a supervisor to make sure that he could refund the Amex voucher so that it would go straight back into my account, he did the refund. Then he waited until the voucher showed up in my account, and then stayed on the line while I booked the new ticket online. Only then did he canx the actual original reservation - just in case something went wrong.

There was a big ^ for him.