Forget it, Jake. It's eBay.
This was the first I’d read about this and I’ve had the very issue you describe with my G4 1.25GHz Mini. On one 1080p monitor, it works fine, but when set to 1080p plugged into my 4K monitor, there’s no image. I have to VNC in, remotely change the settings to something (anything) lower than 1080p and then I’m back in business. Thanks for explaining this issue and ending my frustration at the mystery.Be aware the G4 Mini with the Radeon 9200 has a 135Mhz pixel clock limit over the TMDS encoder for digital displays. Something to do with if the display is coherent or non-coherent.
I ran into this issue a few times with later 1080p TV's.
You just won't get any picture out of them at all if you exceed the 135Mhz pixel clock, however lowering the resolution to 720p works, and you can also monkey around with the settings in SwitchResX.
Unfortunately, display manufactures don't really list if a display is coherent or non-coherent, so it can be luck of the draw for higher res displays.
A good display we know works is I think the 23" Apple DVI LCD made and sold around the time the G4 Mini was new. Other newer higher res single link DVI displays may, or may not work.