So to answer your question.
There is no overlay functionality embedded into LibreTime, you could make the webstream for instance 59:45 seconds long and at the top of the hour schedule a station ID and then resume the webstream. Overlaying station IDs is something that would be interesting to pursue but has not been coded. I believe it would be possible with Liquidsoap but nothing has been done to make this work.
It would be much easier to schedule it to play at 00:00 vs. 59:47 because of the way the scheduling works, you could in theory make the other one happen but this would require limiting all of your shows to 59:47 seconds in length and then adding a station ID show at the end that will always play.
We did add autoloading playlists and intros which can insert tracks at the top of the hour or after your scheduled show but our ability to override scheduling for a station ID at a specific time has not been programmed.