Well, that's not possible with file transfer protocols.
You need a shell access for that.
Actually, you can execute your cd shell command on the Console window:
https://winscp.net/eng/docs/ui_console
You can also setup a custom command, if you need it frequently.
https://winscp.net/eng/docs/custom_command

Would it be possible to allow using server-side $VARIABLES in bookmarks?

A real-life example — on the server I can:

but in WinSCP I can't add a bookmark like this.