The Logitech Media Server is supposed be installed only once in a multiroom environment. The idea of the Squeezebox system is such, that you have one central server (LMS) to control all your players (Squeezelite). The Squeezelite players are always connected to only one server since the server handles the playback. Thus, it is also only possible to sync multiple players on one server, since the server has all the information and control for the players. Here is a picture that details the relationship: https://www.max2play.com/wp-content/uploads/2013/10/max2play_squeezebox-server_concept_-950×720.png
Basically, you only need one LMS and should only have one installed. The players then all show up in the interface with their different names and you easily control them. There really is no advantage to multiple LMS running in the same environment; the Raspberry Pi’s were built to be able to run continuously without powering them down, do not require any significant amount of power (running it through a whole year without interruption would still cost less than 2€) and do not go down easily.