I get the OP frustration with the admin fee. "Admin" fees are the highway robberies of the 21st century, not just in the airline industry. You went 2 pence overdraft? We need a 10£ "admin" fee to be able to deal with it. Cancelling your gym membership? Oh that's going to be a lot of "admin". Etc ad nauseam
I also get the OP's "stick it to the man" feeling. If you're not even going to get a tenner for your hassle, might as well not bother and inconvenience the airline in return (though I doubt it will achieve anything at all).
What I did not get was the reason for cancelling. So you have a guaranteed ticket on a BA flight, non-refundable at that, yet you want to ditch that for a FR flight because that's cheaper and slightly more convenient? Is it worth all the hassle, now that there is no money to save?