OK, KVM extender just came in - and I have to say, I’m impressed! Works plug&play - just inserted it in the HDMI and USB connections, and everything simply worked.
First, I only ran the touchscreen over it, then I got courageous and also connected my MIDI keyboard via USB - and there’s really no noticeable latency or jitter! Feels rock-solid.
I’ll definitely keep everything set up like this for the next couple of weeks and give it a thorough shake, but it looks like I may be on to something. One single Cat6 connection between my on-stage keyboard rig and the Cantabile rack - that’s all it needs to function! Of course, I’d probably keep the USB and HDMI cabling for the “standard” solution with me, but this promises a much more flexible on-stage setup, where I don’t need to keep my Cantabile PC directly around my feet, but can set it up safely at the back of the stage.
Now just a bit more fiddling with the rest of the setup (mic, guitar, in-ear) - I think I’ll wrap that into a nice multi-core and be done with it - two leads from my place to backstage, that’s pretty clean 
Will keep you posted!