Just out of curiosity - why is it difficult to transport the stove from Spain to Austria? I have often carried my gasoline stove (without gasoline, of course) in my checked baggage on flights.

I was flying home from Sevilla with Austrian Airlines, and on their website they state, that it is not allowed to bring any camping stove, neither in hand- nor in checked luggage. I don't know if they actually check/enforce that, but I wanted to be on the safe side.

I ended up shipping the stove separately by post, the ladies in the post office told me that it is also not technically allowed, since it is also shipped by plane, but if there is no remainder of gasoline on it, and I tell them it's a tool or something it would probably be fine.

I figured I'd rather have a package with my stove sent back to Sevilla than miss the flight or have to throw it in the trash at the airport in the worst case.

I have just seen that the complete Lufthansa Group prohibits this in its conditions of carriage.
Google can't find out how long this has been the case.
I would pack the stove, pump and open bottle separately as before.

Clean it and send it home in a parcel ... if it is in parts (disassembled) there will be no problem.

I would advise against taking it on the plane. I missed a flight once for that reason, as per a thorough security check. Which was not fun.

Andy