We will eventually add additional server to the farm. What is the right process of implementing SPLMS in such case?

SPLMS should be deployed to the new server (server with the web application role).

So you will need to deploy our solutions to the new server, run Install-SPApplicationContent​ and restart IIS and SP related services.​
Another variant will be to uninstall LMS solutions and install them again using LMS Suite installation package - you won't need to perform all the operations needed manually.