As has been said, you can do an unlimited number of firearms per 4473 by attaching extra sheets. It all goes under one background check. If two or more of the firearms are handguns, the FFL will also have to file a multiple handgun sale report.
But, if the FFL wants to charge per firearm, that's their business, and you can't force them to do anything else, though you can certainly ask. And since the auction house only wants to use that FFL (they probably have a kickback agreement, btw), you can't even take your business elsewhere.