As I mentioned in comment to kdube, I solved my problem by copying .dll files to my project’s Binaries/Win64 subfolder. It seems that the DLLs I was using were ment for redistribution.
I understand that there is some logic in hiding the compile button when such situation occurs, but it’s not very user friendly. I got stuck with this problem for two days (googling and trial&error attempts) with no useful info from Unreal Editor that would help me solve the problem.
It would be useful to implement some kind of handling of such situations, that would point the user to the source of the problem and help him resolve it. Maybe display a message that compile failed and write some additional info in UE log.