Every individual engine does have it own sound. Not much different, but it's there. Dogs can pick them easily.
I have not been remotely close to your bike, but I'll play the role of the Devils Attorney here:
What if the bike is ok? What if you have become so obsessed by the noise of the engine that you interpret ANY natural noise as a problem?
BMW mass-produce their engines like any other company, and, belive it or not, they do make great efforts to do quality control.
I understand that there has been a new engine installed, and you find there is noise that you interpret as a malfunction.
Soo, let me get the numbers straight: 99.9 % of the LC engines are ok, and occasionally a lemon passes under the radar. Your previous engine might be one of them. BMW replace your engine, and you get a brand new engine installed. And you find this engine faulty as well.
My friend, that is statistically quite a coincidence.
I understand that in your mind there is still a problem, and I can not prove you wrong as such. However, a BMW tech claims that the engine is ok. I know... the tech may be partial, and he might be paid to say so..... . But what if what he says is correct?
You obviously feel like you are banging your head against the wall and feel you get nowhere. But if the tech has examined the engine and given it a green light, you still have the warranty. If the engine does have a problem, your remarks are in the records, and if the problem is real, it will develop further and with a more noticeable fault the bike will be taken care of. (I don't know how BMW UK responds, but BMW takes care of their customers. A friend of mine had a issue with his ESA shock on a K1200GT when it was new. The shock got fixed, but 5 years later an 60 000 km the shock failed again. BMW again replaced the shock since there was a history of issues 5 years back).
If this case was my problem, I would have spoken to the dealer, claimed that even if the tech had given the bike a green light you still felt that something is wrong. I would agree to accept the bike as is for now, and open the door for the fact that the tech is right and that you may be over anxious, but that you would like to come back if your assumption is correct.
For ease of mind, what if you bring the bike to a neutral dealer and have a tech evaluate the bike, without pinpointing what the problem is. If the noise is for real, it should be recognized by others.
I do apologize if I sound harsh now, but I get the feeling that you have painted your self into a corner, and that your pleasure of owning a bike that to most people feels is great is shadowed by an obsession triggered by a problem that has been resolved.
I sincerely hope you find a solution you can live with, as the 1250 is a great bike, and it's a pity that you are not able to enjoy it.
And get a set of earplugs