I've taken a couple of stoves on BA flights, they have always been a bit more reasonable than other airlines about this and other unusual baggage items (not that that's saying a whole lot). Part of the problem is that the person at the ticket counter may have never seen or used a backpacking stove and genuinely doesn't know what's safe and what isn't. Another part of the problem is that some airline check-in staff just like to make life difficult for passengers whenever they can get away with it. Call BA, tell them what you're bringing and ask for their guidelines, then get the name of the person you're talking to. If you can get something in writing, so much the better. If you still don't feel comfortable bringing your stove, just leave it at home and plan on buying something cheap in Europe. FWIW, I have seen at least one airline website that says they will only accept unused stoves in the manufacturer's container as checked baggage. It's a stupid policy, but at least they tell you before you buy a ticket from them.
Way back in the '80s I flew People Express (remember them?) from CA to Maine and back. I was allowed to bring my well-used Svea stove on the eastbound flight, but on the westbound flight it was taken from me because "we just made a new rule, anything that has ever contained flammable liquids is not allowed". They shipped it air freight to me (LOL), and I wrote a nasty, nasty letter to their main office about the way I had been treated. When they went out of business I laughed all day.