The results of your ping means that your laptop can talk to your router, but it cannot talk to the BS5.
On the BS5, go into Settings, then Network Info (not exactly sure what it's called, but it something like this), and note the "Hostname" and "IP address".
If the Hostname is something other than the name you tried above, then try pinging the value found in Hostname.
If this works, then you should be able to get your laptop to talk to the BS5 using that hostname instead of "BeoMaster5".
If this doesn't work, the try pinging the IP address - use the ping command just like before, but provide the IP address instead of the Hostname.
If this works, then you should be able to connect to the BS5 similar to my instructions above, but using the IP address instead of the hostname.
If this doesn't work, check the IP address of your laptop, it should be similar to the IP address of the BS5, except for the last digit. For example, if your laptop's IP address (you can get this by typing "ipconfig" from the command prompt) is 192.1.1.25, the BS5 should be something like 192.1.1.15 - the last number can be pretty much any number except 25 (yes, I'm simplifying). There are exceptions to this, but if you're not on the same IP domain (the 1st few numbers of the IP address), then the router may not understand how to route messages to the BS5. This would happen if the BS5 was previously configured with a static IP address on a different network.
If the IP addresses are meaningfully different, then change the IP address of the BS5. I think you can do this using the service menu. Ideally, you'd want to configure it for dynamic assignment by the router vs. specifying another static address.
You might also verify that the network cable works by swapping the cables around between the laptop and the BS5. Can the laptop still ping the router using the BS5 cable?
If none of this works, then do you have a USB mouse (or mouse/keyboard), and does your SSD have a USB connection? It might be easier to go this path (connect the mouse and SSD to the BS5 and go into the service menu - which allows access to the BS5's underlying windows OS) and copy the files that way vs. troubleshooting network connectivity beyond the above steps.