I'm flying to Hungary, Budapest, in a couple of weeks with rifle and travelling from Gatwick with Easy Jet.
I made enquiries with BA and got the same answer I got when I made enquiries January last year.
£50 for the carriage of rifle, £50 for carriage of other bag with ammunition in and another £50 Admin fee each way !!
They did suggest I could keep the cost down if I carried my ammo in the same case as my rifle, but it would still be £100 each way.
Easy jet, cheap as chips and no bother with the paperwork.