Airline Destroyed My Guitar -- Do I Have Any Recourse?

Hey everyone - earlier this month I was flying from my family's home in LA back to my apt. in London, UK. I'm a working musician so I always bring my guitar on the plane, even though it's in a durable hardshell case I always carry it on with me just as a precaution. (I've done this same LA-London round trip probably half a dozen times and always carry it on, no problem.)

Anyways, as I checked in for my flight at LAX I double-checked with the agent at the desk that there wouldn't be a problem with me bringing it as a carry-on, she said no worries. Now fast forward to everyone boarding the plane, I'm literally walking down the jetbridge to the plane with the guitar in hand and this agent runs up to me and starts trying to grab it. He says there's no space for it, etc., etc., I told him it's quite valuable, and delicate, and I was told can bring it on board. He's extremely adamant it has to go under, says he'll put fragile stickers on it, and it will be handled separately. I eventually concede, thinking ok it's in a hard case, and sure enough when I opened the case at my apartment, the headstock was snapped clean off the neck -- destroyed. It's worth about $800 so it's a big deal for me, and even though I filed a claim with British Airways, they denied any compensation on the grounds that "we ask customers to put anything fragile in their hand baggage" which is EXACTLY what i was trying to do until they barred me from doing so. Do I have any kind of recourse? It's infuriating and I don't want to let them win here lol. Thanks.