Without having witnessed the interaction it is hard to provide judgement on the best course of action. It sounds like the local operator got a bit frustrated and handed out the notice to try and end the matter. If however you think they may have lodged something against your name (which I doubt) then I would write to BA to explain that the notice was inappropriate and handed out in a desperate measure to make up for IT difficulties without any prior warning being given.
As far as the issuance of onward boarding passes is concerned I continue to find it very hit and miss. No problem with BA-BA connections which I had on Saturday, but definitely a problem with AA-BA connection which I had earlier in the month. There the Flagship AA check-in desk at LAX for a LAX-JFK AA flight were unable to issue the boarding pass for my onward flight from JFK-LHR with BA (same ticket, same booking reference). Tried again at the Flagship lounge at LAX and they were also unable to issue the BA boarding pass due to IT difficulties. I was unable to grab it on the BA app either as it did not recognise that we had checked in for the initial AA flight - so definitely some systems issues still in place here which we have experienced on and off for several years!
Just like trying to upgrade an existing booking online that does not involve London as the start or end point has been an IT issue for years. Anything that e.g. starts in LIS, TXL, OSL, DUB, ZRH etc. and connects through to long haul cannot show an upgrade price on the BA website. Get tired of having to phone up my "local BA office" to find out the price and issue the upgrade.
The frustrations here arise I suspect primarily from continued failings in communications between IT systems then compounded by poor human communication.