I assume that you need Slingbox 350 to view the full resolution of the STBs under test. If not, then by far the easiest solution is to continue with the Solo. These are readily available on eBay, etc.
It's unlikely that you could make good use of the old player with 350 or 500. While you might be able to rig an "adapting proxy" and see video, you would still be limited to the resolutions and frame rates supported by the player. IMO, there will not be any enhancements to that product.
Though I've not tested it myself, there are credible posts indicating that once set up, a 350 that does not have access to the public Internet can be viewed, provided that it's directly reachable by TCP. The (web based) player does need to access the Sling servers, so you would set it up with access to both the private network and the Internet. You would make a Slingbox Directory entry pointing to the private IP address and port of the 350. To avoid any security issues, I'd suggest first testing this on a "lab" private network with just one workstation and one 350.
You'll need to know the 15-character admin password, which on 350 / 500 is not normally visible to the end user. To find yours, visit newwatch.slingbox.com and and log into your Sling account (if not automatically logged in). Then, enter https://newwatchsecure.slingbox.com/watch/slingAccounts/account_boxes_js in your browser's address bar. If it offers to save the file, do so and open it in a text editor. If the contents appear in the browser window, copy and paste. You should see data including the passwords for all devices in your directory. If you have multiple devices, please note that the device "displayName" is after the corresponding passwords.
Once you get the above working, you would set up a customized firewall for production use. This would allow the troubleshooting workstation(s) to access only the required Sling servers on the public WAN interface, and only the Slingboxes on the private WAN interface.
If you feel that the above is not sufficiently secure, an alternate approach is for the workstations to have access to only the private network, but use a custom webserver that "spoofs" the responses of the Sling servers, to the minimum extent necessary for the Web player to function.
If none of these solutions are suitable, perhaps you could use the Solos for routine debugging, with an additional non-Slingbox device for auditing video quality.