Replace the HDMI to DisplayPort converter, DisplayPort to Mini DisplayPort cable, and Wacom Link Plus with a Club-3d CAC-1332 which includes a USB 2.0 input that you can connect to the KVM.
If it works, then the display would only work at 4K60. Maybe you can also try 5K39 custom timing.
I think the LG requires RGB input - you'll get weird colors if the HDMI output does YCbCr instead of RGB.
I don't think there are any HDMI to DisplayPort converters yet that can do > 4096 width so the 5K39 idea probably won't work.
https://insights.club-3d.com/thread/hdmi-2-1-to-usb-c-displayport-1-4-alt-mode/
In that case you need a DisplayPort KVM if you want to try 5K39.
It seems to me that the proper way to do KVM with a Thunderbolt display like the LG UltraFine 5K is to create software like Thunderbolt Target Display Mode (used in old 27 inch iMacs that had Thunderbolt 1 or Thunderbolt 2 ports and a 2560x1440 display). You connect two Macs via Thunderbolt 3. The LG UltraFine 5K is connected to one of the Macs. Then the software can alter the tunnelled DisplayPort path between the host Mac and the other Mac. I wonder if the tunnelled PCI path can also be changed? - then you could have the USB features of the display switch between Macs also (but I wouldn't use any storage devices connected to the the LG UltraFine 5K in that case).