Mine wasn't for K1, it was for an EB immigrant visa. But it was just in my handbag, and went through Heathrow security in that. It'll be easy to see on the scanner that it's just documents so I can't see why there would be any issue at all.
I also took another folder of documents in case they were requested at immigration on the US side, again they were just in my handbag and weren't a problem.