It's frustrating that the service menus let us see the network info but not change them

But that's useful info... The printer is looking for the gateway (router) to be at 10.27.0.1. So similar to @debgrub 's suggestion you could reconfigure your router to use that IP.
One option would be to log in to your router and change its network (LAN-side) IP address to 10.27.0.1 subnet 255.255.0.0. Then after rebooting the router, your printer, and your PC, I think that should let the printer find your router, and then the PC to see it (assuming the printer's network interface is actually running, like @debrug wondered). Ofc the change will probably cause all kinds of havoc for all the other devices on your home network so this is just a temporary thing to see if they can communicate and ideally let you access the firmware.
Another option might be to add an advanced routing setting in your router to send any traffic for 10.27.0.1 to your router's current LAN gateway address instead (192.168.0.1 or whatever it is). The advantage would be that it leaves the rest of your network alone so everything else keeps working. I've never done something like that before but maybe someone else can help there with your specific router (mine's an old TP-Link Archer).
Ofc if you can get the router out of the loop entirely and connect ethernet directly to PC like @degrub says should work than that could greatly simply testing.
Does the printer show up at all when connected by USB? Even in the Device Mangler as an unknown device?