Swiss use BA for load control sometimes so its not surprising they have access to there system. Think they handle them or used to at Brum as well.
And BE have code shares with KLM.
But if there are automatic check in stations you can try one of them if it has the airlines logo on it for the second flight.
I turned up at one for a crew positioning flight and it kept on giving me an additional flight. I was due to go to Brum and it was giving me a boarding card from EMA as well.
Just as the terets was kicking in which is quite frequent using those bloody machines I get a phone call from ops giving me the latest plan. I noticed the next bank of them had my third flights airline on it. So gave it a shot and managed to get an Airfrance boarding card out of it.
Also as well some Checkin agents have been doing the job for years and have a black belt in the manual codes in that system that has the blue or black terminal screen to the booking system (forgotten the name of it). They also work multiple airline desks and have login codes for virtually every large operator. So what might be happening is that the checkin desks are setup to be able to check any airline in and they just use un-officially there knowledge to quickly go in and check you in.
Yes this is a nice thing to do.... but.... It gives an inconsistent service level to the punter and can give issues when they travel again when the next agent has only been doing the job for a couple of years and have never used the other airlines systems. The punter then gets upset because they managed it the last time they travelled, and the agent doesn't have a clue what they have done wrong and has no clue how the punter managed it last time so then goes into dealing with a lying tosser pax mode.
So in reality its just creating a PAX expectation problem and generates complaints. So personally I wish they would stop doing it even if they are just being helpful.