is it possible to implement a level change not only by clicking on the server button, but also the client?

You need to call an RPC (remote procedure call) from the client to the server.
This has to be done in an object that is owned by the client (example player controller)

Create a custom event and mark it as run on server. Put your level change logic there hooking up the level reference / name as the functions input.

On ui click have the button contact your player controller and call the new level change function passing in the target level.